COMPUTATIONAL APPROACH TO SOME PROBLEMS IN
ALGEBRAIC ANALYSIS
by
Alberto Damiano
A Dissertation
Submitted to the
Graduate Faculty
of
George Mason University
in Partial Fulfillment of the
the Requirements for the Degree
of
Doctor of Philosophy in Mathematics
Department of Mathematics
Committee:
Dissertation Director
Department Chairman
Dean, College of Arts and Sciences
Date:
Fall 2005
George Mason University
Fairfax, VA
Computational Approach to some Problems in
Algebraic Analysis
A dissertation submitted in partial fulfillment of the requirements for the degree of
Doctor of Philosophy in Mathematics at George Mason University
By
Alberto Damiano
Diploma di Maturità Scientifica
Liceo G. Saccheri, Sanremo, 1997
Laurea in Matematica
Universitá di Genova, 2001
Director: Daniele C. Struppa, Professor
Department of Mathematics
Fall 2005
George Mason University
Fairfax, VA
ii
c 2005 by Alberto Damiano
Copyright °
All Rights Reserved
iii
Dedication
The joy of doing mathematics is a special, weird one. Much as like teaching, it is
incredibly rewarding and terribly frustrating at the same time. Nothing makes me
happier than acquiring new knowledge, and transmitting it to others. And nothing
substitutes the excitement that follows an intuition, be it something that pops in my
own brain or flashes in the eyes of a student. At times, the quest for new results even
took me away from reality and made my vision murky, but the presence of people
around me has allowed me to remain sane. They walked this road with me through
joy and pain and successes and disappointments, and reminded me that life is not
all about numbers, so I would like to honor all of them in the following dedication.
To those who cherish friendship, to those who have a dream and work hard to make
it happen (but know how to be satisfied even when they do not succeed fully), and
those who are proud of who they are but know not to take themselves too seriously,
I dedicate this little dream and achievement of mine.
Daniele surprises me every day more. Not only he is a wonderful advisor, a terrific
person to work with and a passionate mathematician. His energy spreads out form
every single thing he does. Optimistic to the ultimate level, he turns every single
event into a positive experience. He believed in my strengths from the beginning, like
a brother; he guided me through a memorable learning experience, like a mentor and
a teacher; he helped me in those times when I felt lost and wanted to give up, like a
parent. And he still manages to inspire me and motivate me in a unique way. I am
grateful to him for showing me his full personality, allowing me to see how a great
father, husband and friend he is. And if ever I complained about having no time
iv
to do all the things I want, I always shut my negative thoughts because of his great
example as a hard worker. I wish one day I could be to one of my students what he
is to me now. And I feel extremely lucky for being his student, as lucky are all the
people who are next to him.
Vito defines to me what a friend should really be like. His unconditional love,
dedication and care are only some of the things he gives out with no constraints, no
calculations, no doubts or second thoughts. ”Mi casa es tu casa” is not just a saying,
and the fact that I own the keys of his apartment has made me feel more than just
welcome to his place. I felt home. I will miss his Sunday morning breakfasts with
”pan de jamon”, our daily phone calls, and all the fun we had with his endless list
of friends, my ”sister” Tajana and ”Profesaura” Ofelia followed by Rebecca, Nidia,
Edwin, Guillermo, Maria, Angie...
Valerie had me take her to do grocery, laundry, Christmas shopping at ”Potomac
Hells”, browse books and music at Barnes and Nobles, buy modems and Dvd players
at CompUSA, and every time we went out I felt special. Some souls are just meant
to have a good time together, be it spending a memorable vacation in California or
watching our favorite tv shows lying in bed with a soda. Or just working on our
respective laptops without saying absolutely nothing to each other for hours.
My Mom Anna deserves a special mention. How many Italian mothers manage
to see their only child leave for another continent, away from the everyday life and
still are able provide love an support? Not only she patiently waited for my Friday
afternoon phone calls. She also learned how to send and receive emails, how to use
the instant messenger and even how to see me through a web-cam (granted, for that
task some support from my cousins Giulia and Stefano was needed). She knows that
my mood swings and ups and downs are just as tolerable as hers, so she learned how
to cope with that too. And every time I go back home, my room is clean, my piano is
emptied out from all the junk she accumulates during my absence, and a warm slice
v
of ”Sardenaira” is baking in the oven.
Of a different nature, my Father’s help has constantly been there. He is an example
of adaptiveness and versatility. The ”who cares, no big deal” approach to life may
sound a little naive to many, but not to me. Ready to pack a suitcase and leave,
willing to learn a new language, eager to talk to everybody about anything, deep and
personal in his conclusions and with a smart sense of humor; I am glad to say that I
have tried, with some success, to imitate those qualities of him over the past years.
If family is our shelter, our school for life and our place to be our real selves, I
consider myself blessed for having more than just one. My Mom and Renato, my
Dad and Stefania, Vito and Val, Daniele and Lisa, Luisa and Janis, Domenico and
Cheryl, Irene and Fabrizio, I could list many other couples that I can rely on in order
to feel safe. Thanks to these people, ”home” has a special meaning for me now. In
particular, I will be forever in debt to Domenico and Cheryl for picking me up that
day I landed at Dulles for the first time. They started helping me set up a new life
in a new place, a new school and a new environment well before I came to Fairfax.
Their trust, loyalty and friendship were already there, beyond any expectation.
Irene and Fabrizio are yet another amazing couple I am glad I have met on my
path to graduation. Working with them is just so much fun! When they visited
George Mason University over my first year in the States, I had a great time doing
research together, writing papers, proofreading their book, making experiments with
CoCoA, cutting, pasting, trying harder, and starting all over again. All that, of course,
involved drinking a lot of coffee too... I hope that this relationship amongst the three
of us and Daniele will keep us busy for quite a while, as I am looking forward to more
papers, books, results, and pages of calculations.
Ric is simply incredible. His overwhelming and delightful personality makes him
more than ”just a friend”. Or ”just a faculty advisor”. Or ”just a counselor”. None
of the above terms is enough to be applied to him. I thank him for his laughters, for
vi
our chats in the Pride office, for being cool and putting up with my ”eurocentrism”
and my hate for (some) rules. As I type these words on my laptop, I cannot stop
thinking of those days back when, without him, I would have probably gotten on the
first plane to Italy. And now that plane is about to take me home, I realize that I
will miss those little ”pre-phrasing” words he says before he gets to the point, and
those Safe Zone Trainings, and those ”good Ally behavior” lists...
Silvio is a mirror to my soul. If anything is learned in a foreign country, away
from loved ones, where nobody speaks your language and eats what they have told
you is good for you, is your true self. What I need, what I like (or what I don’t), what
I am scared of, and what I can really do to make my own little existence better for
myself, it all came up slowly with his help. Unfortunately there is no Ph.D. in ”self
awareness” disciplines, otherwise this dissertation would be about that and would
have Silvio sign in as my thesis director.
For being the perfect office mate, the perfect person to share a lab and to spend a
lunch break with, and for politely laughing at my ”so cute” Italian accent, I need to
say thanks to Anthony, whose only flaw is probably that of being... already committed
in a relationship with somebody else! I hope he and Dallas will decide to come visit
me soon in Italy, and possibly move to California where I will be more than happy
to go visit them in return.
Indeed, I would like to dedicate this work to all my peers and friends with whom
I have been in touch over the past few years. From just one hour to three years, it
is always worth it to spend time with somebody. Some of them gave me their time
and company since the very first days. Brendan, Lori, Alex, Waverly and Stephen
fall into this category. I say thanks to them as well as to the other members of Pride
Alliance for making me feel accepted and sharing with me some really good moments.
With Liz, Maram, Zac, Alli, Jess, Emily, Kim, no matter what the occasion was, I
always had a good time.
vii
Some others had the advantage of speaking my native language, hence relieving
my effort of cultural adjustment: Domenico, Davide, Mario, and my roommate Justin
in his own way. Grazie!
Some, like Eric and Yami, were incredibly different from me and yet so open and
willing to get to know me that I feel bad, if any regret is allowed here, for letting the
first come too close and keeping the second away from the real me.
On the other side of the ocean, so many people waited, wrote me emails, sent me
text messages and made me feel we were still in touch despite the distance. My three
”Alby’s Angels” Jeorghia, Maria and Stefania are of course on top of my list, but only
because Stefano ”Stezza” and Gabriele ”Gekko” are only two and already have their
own special nicknames. Jeorghia owns the record of most numbers of text messages;
Stefania the one of longest overnight overseas overrated phone calls; Maria wrote the
longest, most dramatic, most ”felt” emails. They all came visit me in Sanremo during
our summer vacations or over the winter break. Every time I saw them, time stood
still and nothing seemed to have changed. As for the guys, they never stopped making
me feel special, be it with some last minute request of translations for Stezza’s last
paper or with Gekko’s Monday’s Extra Help, a pleasant series of emails to brighten
up the beginning of my weeks.
Then come Michela, Paolo, Sabrina, Matteo, Fabrizio, Cristina ”Kappa”, Andrea,
Andrea ”Il Fisiko”, Fabio, Luigi, Leo... I would need to pull out my entire address
book from my PC to name them all. Let me just say this: I am coming back guys!!!!
Some of my special friends were so brave that they jumped on an airplane, flew to
Virginia and overcame the distance. My Father, only few weeks after my first arrival,
was followed by a memorable visit of Jeo and Sabrina several months later. Then my
Mother, my Aunt ”Zia Mimma” and their friend Luci came over to see what it was all
about. I had a blast tour-guiding them and showing them my American life. Kappa
and Massi kept me company shortly afterwards. They really wanted, like Gekko did,
viii
to see D.C. and New York with me. Could I have said ”no” to any of them? I am
Italian, so I love to have guests visit me anytime, anywhere, anyhow. It makes me
happy.
Dedications should always end with a special someone. I am sure that there will
be no jealousy if I thank here the wisest and most unique person of my family. For
being the only one who constantly wrote me letters by hand, and for reminding me
everyday that being able to travel, learn, and meet new people really gives a deeper
meaning to our days, I owe Zia Mimma the biggest hug that words allow.
ix
Acknowledgments
I would like to thank the College of Arts and Sciences, the School of Computational
Sciences and the Department of Mathematics for supporting my studies and my
research at George Mason University, and my parents who let my use their credit
cards to buy my flight tickets and sent me money when I needed some extra cash.
To Lorenzo Robbiano I renew my gratitude for making this experience at George
Mason possible by putting me in touch with Daniele and his group. In addition, I
thank him for always having something funny to say and making me feel at ease in
every situation. I am thankful to him and to the CoCoA Team for trusting me and
believing that I was the right person to ”ship” abroad and work with CoCoA in a
non-standard, non-commutative setting like Computational Algebraic Analysis.
I am particularly grateful to the Office of International Programs and Services for
their incredible work in advising me and the other international students and helping
us overcome the bureaucracy and hidden hassles of immigration. In particular, I have
always appreciated Sandarshi for sharing some of her personal thoughts with me and
caring about my well being more than expected. I am also in debt to the Health
Center and the Counseling Center for taking care of my health during my stay in the
US.
Valerie’s generosity went beyond imagination. I want to thank her once more for
providing me with most of the high-tech devices I now own, including my new laptop
”Ted” with which I typed basically the entire dissertation, and the mp3 player that
entertained me over the whole process (and during my numerous trips around the
world).
x
For their thorough work and their uplifting personalities, I would like to thank
the Dean’s office assistants. In particular, I am grateful to Chrisi for her help in
organizing my first conference, to Gail for accommodating all my requests and to
Heather for always having a smile and something nice to tell me.
Because I feel that it is never enough to mention them, as they still teach me how
to do mathematics, yet another ”thank you” goes to Daniele, Irene and Fabrizio for
their patience, trust and energy in working with me.
The work I present has mostly been the result of team collaboration. Some of
the ideas presented in the following chapters were inspired by a paper, some by a
conversation over a cup of coffee (mostly ”americano”), some others were discussed
over electronic mail. Without aiming for completeness, let me mention some of the
people who participated in this work with me. The introduction has been reviewed
multiple times by Daniele. I owe him a particular attention to presenting things
synthetically, or, as he uses to say, ”from distance”. Chapter 2 would have a different
face if it were not for Irene. Since the first day we met in Milan, she has strived
to help me catch up with the material on linear operators and with the results they
already published before I started working with the group. She was willing to give
up her time just to teach me things that now I really see as the basis of what I know
about algebraic analysis. I think this is priceless. Chapter three is taken from my first
paper, written with Daniele, Irene and Fabrizio jointly. A long way to publication,
but as I already said, a lot of fun! Thanks again to Daniele for giving me the problem
on biregular functions as an exercise for my comprehensive exam and pushing me to
continue the exploration to come up with another paper with him and Irene, who as
always has been a very careful proofreader and coauthor. The chapter on invariant
operators is a collection of all the ideas that we were able to gather in just one
(intense) week of research with V. Souček and J. Bureš at Mason. Six brains together
(Daniele, Irene and Fabrizio were with me as well) can really give birth to unexpected
xi
results. Chapter 6 is inspired by an idea of Frank Sommen, Irene and Daniele. Being
able to transfer it from an experimental paper to the language of CoCoA was my
duty, and I thank Daniele for guiding me over the analysis of the results and for
even sitting down with me to double check the calculations and the code itself. Yet
another evidence that I am very lucky to have him as advisor. Finally, the chapter on
noetherian operators. It is probably one of the most exciting topics I ran into, and
still somewhat mysterious ,which is why it is so interesting. I need to say thanks to
Fabrizio for always reminding me of the analyst perspective, and to all the people like
Bernd Sturmfels, Serkan Hosten, Carlos Berenstein and Leon Ehrenpreis for showing
me their enthusiasm for what I was doing and providing me useful suggestions. And
of course, to Daniele and Irene for writing the paper with me. This dissertation also
contained originally an appendix that in the last versions has been removed to make
the dissertation less heavy, and because it merely consisted of CoCoA code. Although
not visible anymore, it hides the inspiration of somebody as the other chapters do.
Indeed, the person to which I can relate the most when it comes to writing CoCoA code
is Anna Bigatti, who got me interested in this amazing computer algebra tool since I
was a freshmen in Genova. The n-th big ”thank you” goes to Irene for reading several
earlier versions of my dissertation and quickly spotting typos, mistakes and providing
me useful pieces of advice.
I appreciate the cordiality of my committee members, Daniele Struppa, Carlos
Berenstein, Jay Shapiro and Klaus Fischer, and their patience in monitoring my work
since the beginning of my comprehensive exams to the defense of this dissertation.
I would also like to thank David Walnut for his help in joining the Ph.D. program
and the time he dedicated to this, and all the other members of the Department who
showed so much enthusiasm in my enrollment in the new program.
For sending me to places like Greece, Sardinia, Sicily, Spain, South Carolina,
Torino, Austria, I would like to thank again the CAS, the SCS, the Graduate Student
xii
Travel Found at George Mason University, the Politecnico di Torino, the University
of Ferrara, the CoCoA school and Shell Oil, the ECCAD organizers, the University of
Pisa and the RICAM institute of Linz for inviting me to their conferences and for
supporting my expenses. Since traveling and presenting a talk are two of my biggest
passions, being able to experience both has always been a great pleasure. Now, I look
forward to moving to Prague and get to know all the people working on Algebra and
Geometry at Charles University.
Nobody ever complained, even though I was not a music major, that I occasionally
used the baby grand pianos in the Performing Art Department at George Mason to
practice. More than just a hobby, playing music keeps me alive and connected to
a part of reality that the rational mind just cannot grasp, so I need to thank their
flexibility in letting me use the practice rooms.
Thanks to those teachers that reminded me of how much I love to study, and
in particular how much I love to study Mathematics. Evelyn Sander taught a very
interesting course in differential equations and Dynamical Systems; Tim Sauer really
got me into numerical analysis even though I always thought it was not my cup of
tea; Harbir Lamba, with whom I also enjoyed teaching a class in the Spring during
which this dissertation was written, made me feel more ”home” with his European
style of lecturing; Larry Kerschberg gave me and my friend Scott the opportunity to
present our research project in the beautiful scenario of one of the most famous Greek
islands; John Wallin opened my mind on some programming techniques that greatly
changed my approach to writing a code.
As I also found myself ”on the other side” of a classroom, I would like to spend
my last acknowledgment honoring all my students who patiently put up with my
fuzzy ”italianish” names for curves and theorems, with my non anglo-saxon ”4”s and
”7”s, who bore my sometimes tough grading method and still politely pretended to
laugh at my math jokes. Their enthusiastic (and sometimes flattering) feedback on
xiii
my teaching provided me the energy to keep focusing on my research and conclude
this work on time.
xiv
Table of Contents
Page
Abstract . . . . . . . . . . . . .
1 Introduction . . . . . . .
1.1 Motivations . . . . . .
1.2 Goal of the dissertation
1.3 Computational Tools .
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1.4 Structure of the dissertation . . . . . . . . . . . . . . . . . . . . . . .
2 Background on computational algebra and algebraic analysis . . .
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9
2.1
2.2
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4
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Gröbner Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Linear constant coefficient operators . . . . . . . . . . . . . . . . . .
9
16
2.2.1
Syzygies and compatibility conditions . . . . . . . . . . . . . .
18
2.2.2
2.2.3
Free resolutions . . . . . . . . . . . . . . . . . . . . . . . . . .
Ext modules, compact singularities and duality theorems . . .
23
25
2.3
The Cauchy-Fueter operator . . . . . . . . . . . . . . . . . . . . . . .
30
2.4
The Dirac operator . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
Surjectivity theorems and regularity of differential operators . . .
55
3.1
Systems of differential equations . . . . . . . . . . . . . . . . . . . . .
57
3.2
Regular sequences of matrices . . . . . . . . . . . . . . . . . . . . . .
58
3.3
Surjectivity results and some applications . . . . . . . . . . . . . . . .
66
Biregular functions of several quaternionic variables . . . . . . . .
72
4.1
Biregular functions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
72
4.2
Examples of computation in low dimension . . . . . . . . . . . . . . .
74
4.3
Algebraic analysis of the module associated to biregular functions . .
82
Invariant operators and the Cauchy-Fueter complex . . . . . . . .
95
5.1
96
99
Parabolic Geometries . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.1 Invariant first order operators . . . . . . . . . . . . . . . . . .
5.1.2
5.2
Certain invariant second order operators . . . . . . . . . . . . 101
Orbits of the Weyl group in the weight space . . . . . . . . . . . . . . 102
xv
Quaternionic geometry . . . . . . . . . . . . . . . . . . . . . . 102
5.2.2
Invariants for the Cauchy–Fueter operator . . . . . . . . . . . 105
5.3
The construction of the sequence using invariant operators . . . . . . 106
5.4
Equivalence with the algebraic resolution . . . . . . . . . . . . . . . . 110
5.5
Explicit compatibility conditions . . . . . . . . . . . . . . . . . . . . . 113
5.6
6
5.2.1
5.5.1
The complex for 2 operators . . . . . . . . . . . . . . . . . . . 113
5.5.2
The complex for n ≥ 3 operators . . . . . . . . . . . . . . . . 119
Some computations with CoCoA . . . . . . . . . . . . . . . . . . . . . 127
Computation of Dirac syzygies using megaforms . . . . . . . . . . . 132
6.1
The space of megaforms . . . . . . . . . . . . . . . . . . . . . . . . . 133
6.2
Explicit construction of the Dirac complex . . . . . . . . . . . . . . . 135
6.3
6.4
The idea on CoCoA . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
The algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
6.4.1
Experiments with 2 Dirac operators . . . . . . . . . . . . . . . 146
6.4.2
Experiments with 3 Dirac operators . . . . . . . . . . . . . . . 148
6.4.3
Experiments with 4 Dirac operators . . . . . . . . . . . . . . . 155
6.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
7 Noetherian operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
7.1
Fundamental Principle . . . . . . . . . . . . . . . . . . . . . . . . . . 160
7.2
Primary decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . 162
7.2.1
Primary ideals and associated primes . . . . . . . . . . . . . . 163
7.2.2
Existence of an irreducible decomposition . . . . . . . . . . . . 168
7.2.3
An algorithm for primary decomposition . . . . . . . . . . . . 171
7.3
Multiplicity variety . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
7.4
Noetherian operators for a zerodimensional ideal . . . . . . . . . . . . 181
7.4.1
7.4.2
7.5
Closed differential conditions . . . . . . . . . . . . . . . . . . . 182
An algorithm using the Taylor polynomial . . . . . . . . . . . 186
Extension to modules . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
7.5.1 Multiplicity variety for modules . . . . . . . . . . . . . . . . . 191
7.5.2
Noetherian operators for a zerodimensional module . . . . . . 193
7.6
The positive dimensional case . . . . . . . . . . . . . . . . . . . . . . 198
7.7
An optimization of the algorithm . . . . . . . . . . . . . . . . . . . . 210
7.8
Some remarks on complexity . . . . . . . . . . . . . . . . . . . . . . . 215
xvi
8
Future research and open problems . . . . . . . . . . . . . . . . . . . 222
8.1
Analytical problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
8.2
Algebraic problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
Abstract
COMPUTATIONAL APPROACH TO SOME PROBLEMS IN
ALGEBRAIC ANALYSIS
Alberto Damiano, PhD
George Mason University, 2005
Dissertation Director:
This dissertation is the result of my research on the algebraic analysis of linear
constant coefficients differential operators, and its computational aspects. It is the
natural continuation of the work I initiated with my Laurea Thesis [27] where I studied
the computation of the Ext modules using Gröbner Bases and their analytical meaning
for the function theory of kernels of linear constant coefficient partial differential
operators. As the research on Dirac operators developed, new theoretical aspects
have emerged and new relations with computer algebra and Gröbner Basis theory
have been considered. In this dissertation I will provide some background on the
algebraic analysis of such operators, and will then summarize the latest results on
this topic before proceeding to a detailed exposition of my own results. I will devote
particular attention to the use of Gröbner Bases and the experiments conducted with
some computer algebra software packages such as CoCoA [21], Macaulay2 [44] and
Singular [46]. Part of the work is also the CoCoA code I wrote to perform experiments
and to implement the algorithms for the explicit calculation of the mathematical
objects involved in the Fourier analysis of the operators.
Chapter 1: Introduction
1.1
Motivations
In the last few years some mathematicians in the Clifford Analysis community, have
begun to show interest in the application of the new exciting developments in computational algebra to the study of special classes of homogeneous linear constant
coefficient systems of partial differential equations. This program was originally outlined in [68], and has recently found significant expansion in [25]. Basically, one wants
to make effective the results from Algebraic Analysis, which provide significant links
between the global properties of the space of solutions of such system of equations,
and the algebraic properties of the symbols of the associated differential operators.
For example, the removability of compact singularities is related to the vanishing
of some cohomology modules; the description of the compatibility conditions on the
corresponding inhomogeneous systems can be expressed in terms of syzygies; even
the expression of the general solutions through integral formulas is strictly related to
the multiplicity and some other geometric properties of a module. A first example
of a linear constant coefficient differential operator to which this approach can be
fruitfully applied, is the well known Cauchy-Riemann operator
∂
∂
∂
=
+i
∂ z̄
∂x
∂y
1
2
acting on infinitely differentiable functions on R2 . Its kernel is the space of holomorphic functions when we identify R2 with C. Though the classical theory of holomorphic functions does not rely on the algebraic approach we have just described, some
of its more modern variations (e.g. the theory of hyperfunctions as boundary values
of holomorphic functions) are intrinsically linked to this approach. Building on this
new understanding, mathematicians have recently shifted their attention to higher
dimensional operators such as the Cauchy-Fueter operator, the Moisil-Theodorescu
operator, or the more general Dirac operator. To some extent, these operators can
all be considered generalizations of the Cauchy–Riemann operator, providing a more
general definition of ”holomorphicity” in various situations. For example, the kernel
of the Cauchy–Fueter operator defines the space of regular functions on the field of
quaternions H, the kernel of the Dirac operators consists of monogenic functions with
values in a Clifford algebra and so on. The fundamental early work in the study of
linear constant coefficient partial differential operators was carried out independently
in the 1960s by Ehrenpreis and Palamodov (see [35, 60], but also [32]). Among other
things, and in the framework of a fully algebraic treatment of differential equations,
they provided a new interpretation of the Hartogs theorem using Ext modules, and
a general integral formula for the solutions of a linear constant coefficient system
of partial differential equations (the remarkable Ehrenpreis-Palamodov Fundamental
Principle). However, such techniques were hard to use in concrete cases, and had
never been applied to specific systems of differential equations. It is only with the
later work of D. C. Struppa, C. Berenstein, P. Loustaunau and I. Sabadini [1–4, 68]
that such techniques have been reconsidered and applied with the systematic use of
computational algebra and Gröbner Bases. This gave rise to computational algebraic
3
analysis (at least for the constant coefficients case). Computing syzygies, free resolutions, Ext modules, and ultimately noetherian operators are some of the main tools
in the field of computational algebraic analysis. The algorithms that provide the explicit computation of such algebraic objects have been developed thanks to the recent
development of the theory of Gröbner Bases [16]. In particular the birth of various
software packages that allow such computations to be performed with a computer
has been crucial for the growth of computational algebraic analysis. Because of my
particular background and interests, and along the lines of what already done in [25],
I have chosen to adopt the point of view on Gröbner Bases given by [54], as well as
most of its terminology and notation. Consequently, I have made use of CoCoA [21]
as my primary tool for computer applications (this statement needs to be qualified,
since in some cases I have actually used Singular [46] and Macaulay2 [44] that provide
a more complete library of procedures, though much less ”user friendly”).
1.2
Goal of the dissertation
A theorem stating the existence of a mathematical object has an intrinsic beauty in
itself. The Fundamental Principle of Ehrenpreis–Palamodov, for instance, implies the
existence of some differential operators with polynomial coefficients that are used in
an integral-exponential formula for the general solution of a linear constant coefficient
system of partial differential equations: it is a striking result which generalizes the
well known result of Euler on the exponential representation of solutions of ordinary
differential equation, as well as the much more recent representation which Schwartz
gave for mean-periodic functions. On a (very) related topic, but earlier on in the
history of mathematics, Hilbert’s Syzygy Theorem implies the existence of a finite
4
free resolution for any finitely generated module over the polynomial ring; again, this
was a remarkable result when it was first proved. These are just two major examples
of ”existence” theorems, which for years have not yielded to a constructive approach.
My research at George Mason University has focused on the study of results that link
analytic properties of an operator to algebraic properties of the associated module.
My goal has been to find efficient ways to use a computer to calculate the algebraic
objects required by these results, as well as to explore the theory of algebraic analysis
from different points of view by identifying new links between the algebraic nature
of the operator and the analysis of the system of equations. Moving my steps within
these two paths has often required more than a mere use of commands available
through the standard libraries of CoCoA, Singular and Macaulay2. The representation
of the operators as matrices, the calculation of free resolutions in a non commutative
setting, and ultimately the explicit computation of noetherian operators, these are
all actions which require both a deeper understanding of the theory that lies behind
them and, when it comes to implementation of the algorithms on a computer, some
involved programming and experimentation. Experiments then lead to new questions
and may open the space to new conjectures and results, in a virtually infinite circle.
It is still striking to me that, with this approach, a computer algebra package that
performs Gröbner Basis computations can be used for analysis. This is probably
the fact that inspired me to join D.C. Struppa and his coworkers in their research
project. Ultimately, my aim is to present this work in the following chapters, including
theoretical aspects, the algorithms I have developed, with the explicit code written
for each procedure and examples of application, and some experiments I ran using
the software packages I mentioned above.
5
1.3
Computational Tools
Throughout my research work at George Mason, as well as during my previous work
at the University of Genova, the theory of Gröbner Bases has always played an important role. Since it has become more than just a mere theory, particularly with
the work of B. Buchberger [15, 16] and the presentation of its algorithm for the computations of Gröbner Bases, some symbolic computer software packages have been
developed by the mathematical community. CoCoA, Singular and Macaulay2 are just
a few examples. Such tools not only save the researcher from the hassle of manually computing complicated objects such as syzygies, resolutions and cohomology
modules, but quite often, especially when the dimension of the problem is very high,
their use is the only way to actually perform a computation that otherwise would
take an enormous amount of time - and effort - to be carried out. In many cases,
in fact, these computations would be quite impossible if it were not for the use of
appropriate software. A large number of examples I provide in this dissertation fall in
this category. What these packages allow us to do is both to check theoretical results
with explicit examples, and to perform ”experiments” that can be used to derive some
general results.This heuristic approach is definitely not new to many other disciplines,
in particular to computational sciences. However, when it comes to mathematics it
can be considered as a fairly new approach. Sure enough, every mathematician has
some ”working example” that leads him through his research and development of
the theory. However, more complicated and maybe meaningful examples are hard to
explicitly compute by hand. If we consider the case of the computation of the free
6
resolutions for the systems introduced in the next chapter, we can definitely appreciate the power of such computational instruments. For example, the ”simple” case of
3 Cauchy–Fueter operators is actually crucial since it reveals the existence of exceptional syzygies, but it would have been impossible to perform the computation of its
resolution by hand. In conclusion, CoCoA and the other programs have helped us both
in programming a machine to perform some algorithmic procedures arising from the
theory of algebraic analysis (see for example [27] for the computation of Ext modules
and [29] for the computation of noetherian operators) and to conjecture and derive
some facts about the resolutions for the Dirac and Cauchy–Fueter operators. This
dissertation will present all the cases when this has been particularly true, showing
both the outputs I got using CoCoA and the way they have influenced and sometimes
determined my research. All the algorithms I coded will be made available through
our web page [28] with explanations, examples, performance analysis and updates.
Finally, a comprehensive package including all the procedures presented in the dissertation has been created specifically for CoCoA and will be included in the latest
version of the software. The latest version of the package is available through our
CoAlA webpage www.tlc185.com/coala
1.4
Structure of the dissertation
The second chapter of this dissertation presents an overview of the main aspects of
algebraic analysis of linear constant coefficients partial differential operators. The
focus will be on the relationship with computer algebra to introduce what is now
known as computational algebraic analysis as it is beautifully portrayed in [25]. This
is an introductory chapter, which will include only few main results. The attention
7
will be devoted mostly to the use of CoCoA to calculate syzygies, free resolutions and
Ext modules for some classical linear systems: the Cauchy-Riemann system defining
holomorphic functions, the Cauchy-Fueter system defining regular functions on H and
the more general Dirac system whose space of solution are called monogenic functions.
Chapter 3 presents a first original result on the surjectivity of some particular differential operators and the possibility of describing the entire free resolution of a module
associated to a system of differential equation using the block structure of the symbol
matrix.
Chapter 4 includes an example of application of such techniques to a non trivial
generalization of the Cauchy-Fueter system. I here use the tools and ideas discussed
in the previous chapter, plus an explicit calculation of a Gröbner Basis to show how
algebraic analysis techniques can be applied to the case of the so-called biregular
functions of (several) quaternionic variables [30]. Of a completely different nature
is Chapter 5. Here I will follow an apparently different path, using nothing but
representation theory and invariant operators, to describe (again) the resolution of
the Cauchy-Fueter operator. This approach has captured our attention thanks to
the work of V. Souček and J. Bureš and some related results presented for instance
in [5, 51]. Although such method does not utilize any Gröbner Basis techniques,
the resolution obtained is equivalent to the one we obtain through the algebraic
analysis of the system. A result stating the equivalence of the two complexes then
gives a surprising and meaningful bridge between the two theories. I present it in
Theorem 5.2. Our research on this topic also explored the possibility of using complex
coordinates instead of real ones for the Cauchy-Fueter system, as suggested by the
representation theory. As yet another positive surprise, this alternative can be coded
8
on CoCoA as well, producing an equivalent resolution for the system.
In the case of regular or biregular functions of several quaternionic variables, the real
dimension 4 of H is relatively disadvantageous for the Gröbner Basis computation.
Moreover, when it comes to monogenic functions with values in the Clifford algebra
Cn the real dimension prevents us from constructing the explicit resolution even for
small values of n. An alternative approach that makes no use of the real components
has been envisioned and presented in [62] and is briefly summarized in Chapter 6. My
goal here has been to try to implement such ideas on CoCoA, overcoming the obstacles
that the non-commutativity of Cn implies in this case. I show that, even though the
complexity of the new approach is still exponential, we can use these techniques to
compute a (not minimal) resolution of the Dirac system.
Chapter 7 is an extensive and thorough presentation of my research on the explicit
construction of noetherian operators. It includes an introductory, and pretty much
self-contained section on an important algebraic tool needed for the construction: the
primary decomposition of a polynomials ideal, and its computational counterpart.
Thus I present the results of our paper [29] as well as some further development and
optimizations conducted after its submission.
The last chapter contains a list of open problems and possible directions for future
research. The CoCoA code I wrote for each algorithm presented in the dissertation
and for the procedures used for my examples and experiments is included in the new
CoCoA package coala.cpkg that will be available on the next version of CoCoA and
can also be downloaded at our webpage [28].
Chapter 2: Background on computational algebra
and algebraic analysis
2.1
Gröbner Bases
I will fix here some notation that I will adopt throughout the dissertation, and give
an overview of the main concepts from the theory of Gröbner Bases. On this topic,
I will follow the point of view of [54] to which I refer the reader for all the proofs
and for more details. We will work in the ring R = C[x1 , . . . , xn ] of polynomials
in n variables with complex coefficients; we will think of R as the ring of symbols
for the differential operators we are studying. Even though we consider differential
operators with constant coefficients, we will sometime need to work in the Weyl
Algebra C[x1 , . . . , xn , ∂x1 , . . . , ∂xn ]. The symbol ∂x is a shortcut for
∂
.
∂x
Using the notation introduced in [54], we will denote the monoid of power products
in R by Tn and the module monoid of power products in Rs by
Tn he1 , . . . , es i = {tei |t ∈ Tn , i = 1, . . . , s}
where ei is the i-th element of the canonical basis of Rs . A term ordering σ on Tn
(resp. a module term ordering Tn he1 , . . . , es i) is a total ordering on power products
with the following two properties:
1) if t1 >σ t2 and t ∈ Tn then t · t1 >σ t · t2 ;
2) if t ∈ Tn (resp. t ∈ Tn he1 , . . . , es i) and s ∈ Tn then s · t >σ t. Given a term
9
10
ordering σ on Tn it is possible to extend such ordering to a module term ordering τ
on Tn he1 , . . . , es i so that the module term ordering obtained is compatible with σ, i.e.
if t1 >σ t2 then t1 ei >τ t2 ei for every choice of ei in the canonical basis of Rs . This
can be done essentially in two ways. In the first case the ordering τ is called ToPos
and is defined as follows:
t1 ei >τ t2 ej
⇐⇒
t1 >σ t2 or (t1 = t2 and i < j)
which means that we are ordering first terms and then positions; in the second case
we do exactly the opposite, ordering positions and then terms (PosTo) as follows:
t1 ei >τ t2 ej
⇐⇒
i < j or (i = j and t1 >σ t2 ).
The leading term ideal (resp. the leading term module) associated to I (resp. M )
with respect to σ will be indicated by
LTσ (I) = {LTσ (f )|f ∈ I}.
More in general, the leading term ideal associated to a subset G of R will be written
as LTσ (G) = {LTσ (f )|f ∈ G}. Note that LTσ (G) = LTσ (I) if and only if the set G
is a Gröbner Basis for the ideal I (the same holds for modules), this being the main
characterization of a Gröbner Basis.
The algorithm which associates to an ideal I of R (resp. a submodule M of
Rs ) its Gröbner Basis Gσ (I) (resp. Gσ (M )) is the core algorithm of the theory of
Gröbner Bases and can be found for example in [54], theorem 2.5.5. Another key
tool in computational algebra is the division algorithm (see again [54], theorem 1.6.4)
which can be performed to generate the remainder of a polynomial (resp. vector)
11
with respect to a set of generators of I (resp. M ). Note that the remainder of a
polynomial depends on the set of generators chosen for I (in fact, it even depends on
their order). In particular it is possible for a polynomial to belong to an ideal and
yet to have a remainder different from zero with respect to some set of generators.
The fundamental property of Gröbner Bases is that such a remainder is zero if and
only if the polynomial belongs to the ideal. For this reason the remainder calculated
with respect to a Gröbner Basis is called the normal form of a polynomial. Therefore,
computing the normal form of a polynomial (resp. vector) with respect to a Gröbner
Basis is useful to test whether the element belongs to I (resp. M ). We will use this
fact, for example, for the computation of noetherian operators in Chapter 7, since
such operators constitute an alternative membership test for the ideal I (resp. module
M ) as well.
Given a polynomial f ∈ I and a term ordering σ, we will denote by NF(f ) the
normal form of f with respect to the σ–Gröbner Basis of I (the same notation is used
for modules). An equivalent way to compute a remainder is using rewrite rules (see
[54] section 2.2). Given a polynomial g ∈ R, we say that a polynomial f1 rewrites to
g
g
f2 with respect to the rewrite rule −→ (and this is indicated by f1 −→ f2 ) if there
exists a monomial m in R such that f2 = f1 − mg and LTσ (mg) is not in the support
of f2 . This is also called a one-step reduction. We can rewrite a polynomial using
a set of elements G = {g1 , . . . , gs } by performing a one-step reduction with each of
G
the gi ’s, in that order. We will denote by −→ the transitive closure of the relations
g1
gs
−→, . . . , −→. This relation is called rewrite relation or rewrite rule. By applying a
sequence of one-step reductions to a polynomial f using the elements in G we then
12
obtain a remainder of f with respect to {g1 , . . . , gs }. In particular if G is a Gröbner
G
Basis we have that f rewrites to its normal form, i.e. f −→ NF(f ).
Let me now introduce three objects that play a crucial role in algebraic analysis:
syzygies, free resolutions and cohomology modules. They are strictly related to each
other. Given a s-uple of elements f = (f1 , . . . , fs ) in an R-module M , we say that
the s-uple a = (a1 , . . . , as ) ∈ Rs is a syzygy for f if
a1 f1 + · · · + as fs = 0.
The set of all syzygies of f is a submodule of Rs which is finitely generated. It is
often denoted by Syz(f1 , . . . , fs ) or, if the fi are generators of M , by Syz(M ). The
explicit construction of the syzygies is possible and relies on the use of a Gröbner
Basis for the module generated by the fi and on the division algorithm. Suppose we
have calculated the syzygies of f and denote them by g1 , . . . , gr . We can imagine to
iterate this construction and get the generators for Syz(g1 , . . . , gr ), and so on. At each
step, a new finitely generated module is obtained. It is spanned by the syzygies of
the generators of the previous module. Since the syzygies calculated at each step are
nothing but kernel of the polynomial map described by the generators of the module
obtained at the previous step, we can collect all the modules and maps in a complex
ϕn
ϕ2
ϕ1
ϕ0
F : · · · → Fn −→ . . . −→ F1 −→ F0 −→ M → 0.
(2.1)
This complex is exact by construction. Each module Fn is a free module Rrn , where
r1 = r and r0 = s. The last map is given by ϕ0 (ei ) = fi , where ei is the canonical
13
basis element of F0 = Rs . Sometimes we just consider the complex
ϕn
ϕ2
ϕ1
F : · · · → Fn −→ . . . −→ F1 −→ F0 → 0
(2.2)
with the extra assumption that M = Coker(ϕ1 ). The complex (2.2) is called a free
resolution for the module M . The nature of the module M is encapsulated in its
free resolution. Although complex (2.2) is far from being unique, as each set of
generators for the syzygies can be chosen in different ways, in the graded case we
can consider a notion of minimality for free resolutions. When the syzygies that
constitute the image of the maps in the complex are chosen to be minimal among all
the possible sets of generators, then the resolution is said to be minimal. This has a
nice characterization using a matrix representation of the maps. Since the map ϕi is a
polynomial map between free modules, we can think of it as a matrix whose columns
are the generators of Im(ϕi ). The minimality of the free resolution then corresponds
to the fact that in each matrix the nonzero entries have degree at least one. Two
minimal free resolutions of the same module are isomorphic as complexes [36]. In
particular, they have the same length and the free modules have the same ranks,
and these numbers are minimal among all the possible choices of the free resolution.
Most of the numerical invariants associated to M then arise from this uniqueness
argument. The ranks of the free modules Fi are called Betti numbers. They contain,
together with the degrees of the maps ϕ1 all the information on the Hilbert series
of M (see for example [37]), the Krull dimension, the depth and the homological
properties of M . Let us now talk about the graded version of (2.2). We will endow
the ring of polynomial R with the natural grading where each variable has weight
one. Consequently, a free module Rs inherits a natural Z-grading which respects the
14
structure of module (i.e. the canonical basis elements are homogeneous elements of
degree zero). It is possible to consider graded submodules M of Rs and construct a
free resolution with homogeneous maps. Suppose that the degree of the generator fi
of M is an integer di > 0. Then we can construct the syzygies in the free module
R(−d1 ) ⊕ · · · ⊕ R(−ds ),
(2.3)
where the shift R(−di ) means that the basis element ei has degree di . This way,
the first map of the free resolution has the nice property of being homogeneous of
degree zero, i.e. it maps homogeneous elements to homogeneous elements of the
same degree. Iterating this procedure as above, we obtain the so called graded free
resolution of M. It is in this graded setting that it actually makes sense to consider a
graded minimal free resolution, since the presence of homogeneous generators and the
use of Nakayama’s lemma (see [57]) make it possible to choose a system of minimal
generators so that the Betti numbers are well defined.
From what I have introduced so far, the construction of a free resolution seems to
be possible given that we have an effective algorithm for the computation of syzygies,
but I have not said anything about the finiteness of a resolution. In order to assure
that the procedure that builds a free resolution for an R-module terminates, we have
to show that there exist at least a resolution of finite length. This is assured by the
famous:
Theorem 2.1 (Hilbert Syzygy Theorem). Every finitely generated (graded) R-module
has a finite (graded) resolution of length m ≤ n, in which all modules are finitely
generated (graded) free modules.
15
In conclusion, to each (graded) finitely generated module we can associate a minimal (graded) free resolution that is unique modulo isomorphisms. The ranks of the
graded free modules are called graded Betti numbers of M . Another important invariant that the free resolution makes available to us is given by its cohomology modules,
or Ext modules. For a more complete treatment of this subject from the computational point of view see [27]. Here I will just state the definition. If we consider the
dual of the complex (2.2), i.e. the complex obtained by applying the controvariant
functor Hom(−, R) to the free resolution, we obtain
Hom(F, R) : 0 → Hom(F0 , R)
Hom(ϕ1 ,R)
−→
Hom(F1 , R)
Hom(ϕ2 ,R)
−→
....
(2.4)
By definition of Hom functor [54], the map Hom(ϕi , R) is simply the transposed of
the map ϕi , and the free module Hom(Fi , R) is in fact isomorphic to Fi . So, more
explicitly, the dual complex (2.4) is
ϕt
ϕt
ϕt
ϕt
3
n
2
1
. . . −→
Fn −→ . . .
F2 −→
F1 −→
F ∗ : 0 → F0 −→
(2.5)
Note that this is no longer an exact complex. The non-exactness of (2.5) is measured
by the modules
ExtiR (M, R) :=
Ker(ϕti+1 )
.
Im(ϕti )
The construction of such modules is another important landmark in the theory of
Gröbner Bases. In particular algebraic analysts are interested in the vanishing of
such modules.
16
2.2
Linear constant coefficient operators
The first major mathematician who used the term ”algebraic analysis” was probably
Euler, in connection with his important work on general solutions to linear ordinary
differential equations with constant coefficients, [38]. Currently, the term ”algebraic
analysis” refers to the work of the Japanese school of Kyoto (Sato, Kashiwara, Kawai
and their coworkers) who founded and developed methods to analyze algebraically
systems of linear partial differential equations with real analytic coefficients. The first
to develop the algebraic techniques of the Japanese school, in addition to some early
work of Malgrange and Martineau, were Ehrenpreis [35] and Palamodov [60] in the
1960s. However, their results were almost never applied to specific systems of equation. Instead, they were used to provide some general results on the nullsolutions of
differential operators with constant coefficients, like the new interpretation of Hartogs
theorem [34] and the Ehrenpreis-Palamodov Fundamental Principle, [33, 60]. In this
section I will focus on some examples of linear constant coefficient differential operators that have been studied using algebraic techniques and Gröbner Bases. Rather
than discussing general results, I prefer to show some applications to see how the techniques can be applied in practice. Following the notation and terminology adopted
in [25], let us briefly introduce the concept of linear constant coefficient partial differential operator. Given a polynomial p in R, we can obtain a differential operator by
replacing the variable xi with the derivative −i∂xi . Strictly speaking, one would have
to consider different names for the variables and their dual, not to generate confusion.
I will not adopt this convention. Denoting by D = (−i∂x1 , . . . , −i∂xn ) the n-tuple of
derivatives, the differential operator associated to p can then be referred to as p(D).
17
Such operator represents the differential equation
p(D)f = 0
where f is chosen in a suitable sheaf of generalized solutions, such as infinitely differentiable functions E = C ∞ (Ω) on an open convex set Ω ⊆ Rn , holomorphic functions
O(Ω), distributions D or hyperfunctions B (see [25] for the definition of those spaces).
This was the case initially studied by Euler. He proved that it is possible to represent
the solutions to a single ODE via the zeroes of the polynomial p (I will talk about
this as the starting point for the construction of the Noetherian operators in Chapter
7). Roughly speaking, we can say that algebraic analysis is a generalization of this
method.
More in general, if P is a matrix of polynomials P = [Pij ], the differential operator
associated to P will be the matrix P (D) = [Pij (D)]. This operator represents a system
of linear constant coefficient partial differential equations
P (D) · f = 0
for a column vector f of generalized functions, where the product represents the usual
matrix multiplication. The particular case of a column matrix P = (p1 , . . . , ps )t can
be seen as the case of s differential equations for a scalar function f . The case of
a single ordinary differential equation then corresponds to n = 1 and s = 1. The
matrix P is sometimes referred to as the symbol of the operator P (D). Working on
spaces of functions in which a suitable Fourier transform can be defined, we can define
P (D) to be the conjugate of the multiplication operator P . It is then evident that
the properties of the operator P (D) are tightly related to those of the matrix P .
18
Suppose that P is a r1 × r0 matrix of polynomials in R. The algebraic object that
plays the most important role in algebraic analysis is the module
M := Coker(P t ) = Rr0 /P t Rr1 = Rr0 /hP t i.
It is a finitely generated R-module. It is the quotient of the free module Rr0 with
the submodule generated by the rows of the matrix P . We can then describe the
module M using syzygies, free resolutions and cohomology modules. Here is where
the tools of computational algebra introduced in section 2.1 come into the picture.
The module M has a minimal finite free resolution of the form
Prt
t
Pm−1
Pt
1
0 −→ Rrm −→ Rrm−1 −→ . . . −→
Rr1 −→ Rr0 −→ M −→ 0
(2.6)
whose dual is the complex
P
P
Pm−1
1
0 −→ Rr0 −→ Rr1 −→
. . . −→ Rrm−1 −→ Rrm −→ 0.
(2.7)
Question 2.1. What kind of analytical information on the system can we obtain
once we have calculated a resolution like (2.7)?
The next paragraphs will explore the answer to this question.
2.2.1
Syzygies and compatibility conditions
Let us look at the matrix P1 appearing in the complex (2.7). We know that, from
the algebraic point of view, P1t represents the syzygy module of hP t i. But what is its
analytical counterpart? For the sake of simplicity let us suppose that P is a column
19
of 3 polynomials


p1
P =  p2 
p3
representing the homogeneous system of equations p1 (D)f = p2 (D)f = p3 (D)f = 0.
Let f, g1 , g2 , g3 be differentiable functions and consider the inhomogeneous system

 p1 (D)f = g1
p2 (D)f = g2

p3 (D)f = g3
The compatibility conditions of the system are a collection of differential equations
q1 (D)g1 + q2 (D)g2 + q3 (D)g3 = 0
that the functions gi have to satisfy in order for the system to admit a solution. These
correspond exactly to the relations
q1 p1 + q2 p2 + q3 p3 = 0
satisfied by the polynomials pi . In other words, they correspond to the syzygies
of (p1 , p2 , p3 ). This is due to the fact that the operators p1 (D), p2 (D), p3 (D) ∈
C[∂x1 , . . . , ∂xn ] have the same algebraic properties of their symbols, granted that the
commutative property
∂xi ∂xj = ∂xj ∂xi
holds for all i, j = 1 . . . n. More in general, if we have a matrix operator P (D) and
we consider the inhomogeneous system
P (D)f = g,
(2.8)
20
the compatibility conditions will be given by the syzygies of the matrix P t . Equivalently, the datum g has to be in the kernel of the operator P1 (D), so that we can
represent this condition with the new homogeneous system
P1 (D)g = 0.
(2.9)
With the use of sheaf terminology, this can be specified in a theorem (see [33] for a
proof of the case of differentiable functions and distributions).
Theorem 2.2. Let S be one of the sheaves of generalized functions E, O, B or D.
The system (2.8) has a solution f on S(U )r0 on a convex open set U , if and only if
g ∈ S(U )r1 satisfies the compatibility condition P1 (D)g = 0.
Let us look at a classical example: the Cauchy-Riemann operator acting on differentiable functions of several complex variables.
Example 2.1. Let zj = xj + iyj where i2 = −1 and xj , yj are real variables. Define
the Cauchy-Riemann operator
∂z¯j = ∂xj + i∂yj ,
j = 1, . . . , n.
The solutions of the system given by ∂z¯j f (z1 , z2 ) = 0 on an open convex of Cn , are
precisely the holomorphic functions on that open set. If we want to represent the
system using a matrix P , we need to find the corresponding equations for the real
valued functions f1 , f2 , where f = f1 +if2 . This system of equations is the well-known
Cauchy-Riemann system that characterizes the holomorphicity of a function f . The
21
following is the system in the case n = 2.

∂x1 f1 − ∂y1 f2 = 0








 ∂y1 f1 + ∂x1 f2 = 0


∂x2 f1 − ∂y2 f2 = 0







∂y2 f1 + ∂x2 f2 = 0
(2.10)
Note that the factor −i is usually omitted given that we work with homogeneous
polynomials. The symbol associated to this system is then the matrix




x1 −y1
Q
1
 y1
x1 
 
,
P =
 x2 −y2  =
Q2
y2
x2
where Qi =
¡xi
−yi
yi xi
¢
. Basically, we are switching from a representation of the operator
as acting on functions f : C2 −→ C to looking at the corresponding operator acting
on real valued functions f : R4 −→ R2 . The module M = hP t i has four generators
(the rows of P). We can use CoCoA to find its first syzygies, using the command
SyzOfGens:
Use R::=Q[x[1..2],y[1..2]];
P:=Mat[
[x[1],-y[1]],
[y[1],x[1]],
[x[2],-y[2]],
[y[2],x[2]]];
22
M:=Module(P); -- in CoCoA a module associated to a matrix
-- is generated by its rows
S:=SyzOfGens(M,1);
Mat(S); -- this is P1
Mat[
[x[2], -y[2], -x[1], y[1]],
[y[2], x[2], -y[1], -x[1]]
]
------------------------------As we immediately see, the matrix we obtain consists of the two blocks [Q2 , −Q1 ], so
the compatibility condition of the system
½
∂z¯1 f = g1
∂z¯2 f = g2
is expressible in term of Cauchy-Riemann operators as ∂z¯2 g1 = ∂z¯1 g2 . This is a well
known result from the theory of holomorphic functions in several variables.
Remark 2.1. The compatibility conditions for the Cauchy-Riemann system, even in
the general case of n operators are given by
∂z¯i gj = ∂z¯j gi ,
i, j = 1, . . . , m i 6= j.
(2.11)
It is quite remarkable that this purely algebraic approach allows to obtain such a
result without the use of any analytical arguments. The conditions (2.11) come from
the syzygies of the variables (z1 , . . . , zn ). Exactly like these variables, the n symbols
23
associated to the Cauchy-Riemann operators commute because of the relations
∂z¯i ∂z¯j = ∂z¯j ∂z¯i ,
plus, they form a regular sequence. We will see in Chapter 3 that this situation
reflects exactly the property of having ”natural” syzygies of the type (2.11).
2.2.2
Free resolutions
Since we obtain a free resolution for M by iterating the construction of syzygies, each
matrix Pi appearing in the complex (2.7) provides the compatibility conditions for
the inhomogeneous system given by
Pi (D)gi = gi+1 ,
i∈N
where the vector functions gi are chosen in the same sheaf of f = g0 . When can
rephrase this by saying that the free resolution of M contains all the compatibility
conditions, at large, of the original system of equations. Alternatively, we say that the
free resolution is a measure of the compatibility of the inhomogeneous system (2.8).
Let us have a look again at the case of the Cauchy-Riemann operator, performing
some experiments with the use of the computer.
Example 2.2. When working with holomorphic functions of just 2 complex variables,
we see that the resolution ends immediately after the first syzygy map is found.
Continuing from example 2.1 and using CoCoA we get
Res(R^2/M); 0 --> R^2(-2) --> R^4(-1) --> R^2
-------------------------------
24
which means that the free resolution of the quotient is given by
Pt
Pt
1
0 −→ R2 (−2) −→
R4 (−1) −→ R2 −→ M −→ 0
where P and P1 are defined as in section 2.2.1. The only compatibility for the CauchyRiemann system in two variables is then given by the first syzygy map. The resolution
also happen to be linear, as the change in the grading is ”shifted up” by a factor of
one at each step. This means that the generators of all maps have degree one. There
is virtually no difference between this case and the case n > 2. The nature of the
complex in the case of n variables is the same [25]: the Betti numbers and the maps
involved in the resolution follow the same pattern of the so called Koszul complex for
n variables (z1 , . . . , zn ) in C[z1 , . . . , zn ]. (Remember that the Koszul complex is the
free resolution associated to an ideal generated by a regular sequence, see [57] for the
definition and [17] for a generalization to arbitrary modules). Only, we have to take
into account the real dimension 2 of C:
t
Pn−1
t
Pn−2
Pt
Pt
1
R2( 1 ) (−1) −→ R2 −→ 0.
0 −→ R2 (−n) −→ R2(n−1) (−n + 1) −→ . . . −→
n
n
(2.12)
The complex is linear; the maps can all be described in terms of the Cauchy-Riemann
operators ∂z¯i as they just come from the maps of the Koszul complex which are written
in terms of the complex variables zi ; finally, the last map is the transpose of the first
one, i.e.
Pn−1 = P t .
All the properties listed in the previous example are somehow peculiar for this
operator. What is interesting is then to compare this operator and its resolution
25
to other linear differential operators. In the sections and chapters to follow we will
study differential operators for which some of the properties now described hold as
well, but some others fail to be true. For example, the fact that it is possible to
describe the maps purely in terms of the original operators applies also to Dirac
operators (see section 2.4) but only when the dimension of the underlying space is
”high enough”. Quaternions do not satisfy this property in general since H has real
dimension 4. Another pattern that we do not observe in the quaternionic case is
the symmetry of the complex. In particular if the last map is described in terms of
some known differential operators (possibly the same as the original operators forming
P (D)), then a cohomological study of the resolution, that I will briefly describe in
the next section, allows to derive a description of generalized functionals. For the
Cauchy-Riemann operator, since the last map is the transpose of the first and the
dual complex is actually isomorphic to the free resolution itself, we obtain that the
space of hyperfunctions actually coincide with the space of holomorphic functions
itself. However, this is not true for other operators and makes the description of the
functionals even more interesting.
2.2.3
Ext modules, compact singularities and duality theorems
I anticipated many times before this section that the cohomology modules are strictly
related to some analytical properties of the nullsolutions of linear constant coefficient
partial differential operators. I will describe here two important results that constitute
yet another bridge between the algebraic and the analytical nature of the operator.
The first one has to do with the removability of compact singularities from the domain
26
of the functions. It is a well known fact that a function of one complex variable
can have singularities in its domain. In 1906 Hartogs proved that if a function is
holomorphic on an open subset of Cn , with n > 1, it is so ”rigid” that its domain
cannot have compact singularities. More precisely, this property is known as Hartogs’
phenomenon:
Theorem 2.3 (Hartogs’ phenomenon for Holomorphic functions). Let Ω be an open
set of Cn , n > 1 and let K ⊆ Ω be a compact set such that Ω \ K is connected. Let f
be a holomorphic function on Ω \ K with values in C. Then it is possible to uniquely
extend f to a function fe : Ω −→ C such that fe is holomorphic on Ω.
Hartogs proof relied on purely analytical arguments. It is only with the later
work of Palamodov and Ehrenpreis that it became evident that a different proof
could be given in the general setting of linear constant coefficient operators, using
only algebraic techniques:
Theorem 2.4. Let R be a polynomial ring, let P ∈ Mat(R) be the symbol of a linear
constant coefficient partial differential operator. Denote by M the cokernel of the
map defined by P and consider a differentiable solution f to the system P (D)f = 0.
Suppose that f is defined on an open set Ω \ K where K is a compact set in Ω. Then
it is possible to extend f to a function fe such that P (D)fe = 0 on Ω if and only if
Ext1R (M, R) = 0.
Checking if a Hartogs type of phenomenon holds for the sheaf of solutions of a
particular system of constant coefficient PDEs becomes then a matter of being able
to compute the first cohomology module. On CoCoA this can be easily done by using
27
the package $contrib/ext 1 . The command to use is
Ext.Ext(R^S/M,R^1,1);
assuming that M is the module generated by the rows of P , S is the number of components of M and R is the working ring. This will present Ext1R (M, R) as a quotient, and
will return just zero if it is the zero module. Singular has an alternative command:
IsZeroExt_r(1,M);
whose output is the boolean (TRUE or FALSE) corresponding to the vanishing of the
first cohomology module. An alternative way to establish the vanishing of the first
cohomology module comes from the following proposition (see [35] for a proof):
Proposition 2.1. Let M be the R-module generated by the rows of the polynomial
matrix P . Suppose that P is of maximal rank. Then Ext1R (M, R) = 0 if and only if
the maximal minors of P are relatively prime.
The actual calculation of the maximal minors of P is in general computationally
expensive, especially for large matrices. This characterization of the vanishing of Ext
is then useful only in low dimension cases. One can perform it with CoCoA, provided
that P is the matrix representing the operator, by entering the line
GCD(Minors(1,P))=1;
In case the response is positive, this means that the first Ext module is zero.
Like the function IsZeroExt r used in Singular, this avoids the actual computation
and presentation of the module, which is not needed if we just want to deduce the
removability property.
1
CoCoA versions 4.3 and higher include the Ext command by default.
28
Example 2.3. Consider again the system associated to two Cauchy-Riemann operators as defined in example 2.1. Our goal is now to prove with CoCoA that the Hartogs
phenomenon holds for holomorphic functions of two complex variables. We proceed
as follows:
Ext.Ext(R^2/Module(P),R^1,1);
Record(
MOD = Module([0]),
SHIFTS = [0]
)
------------------------------Vanishing of the first cohomology has then been experimentally confirmed. As an
alternative, let the machine check that the minors of P are coprime:
GCD(Minors(1,P))=1;
TRUE
------------------------------Given a sheaf of generalized functions S, we indicate with S P the sheaf of solutions
defined by Γ(U, |S P ) := Ker(P (D)|U ). Then a famous result (see [25], Theorem 2.1.3)
states that the algebraic complex (2.7) can be translated into a complex of sheaves
P (D)
P1 (D)
0 −→ S P −→ S r0 −→ S r1 −→ . . . −→ S rm −→ 0.
(2.13)
The algebraic modules ExtiR (M, R) correspond to the local cohomology groups
i
HK
(Ω, S P ) with coefficients in the sheaf S P . The consequence of this is that it is
29
possible to describe other properties of removability of ”higher order” singularities by
means of the vanishing of higher order cohomology modules [50]. I will not describe
the kind of singularities involved in such results since this would go beyond the goal
of this overview. I refer as always the interested reader to [25] for more on this aspect.
Another very significative property associated to Ext modules and the maps in
the complex is the description of the dual of S P . Let us introduce some more precise
notation. Again this relies on the correspondence between the Ext modules and the
cohomologies of (2.13). Let us set consider, for the sake of simplicity, the sheaf E
of infinitely differentiable functions on Rn . The following is a classical theorem of
duality of cohomology modules, see for example [50].
Theorem 2.5. Let K be a compact set in Rn . Let E be the sheaf of differentiable
functions on Rn . Let M be the module associated to a differential operator P (D) and
let m < n. Suppose that
ExtjR (M, R) = 0,
f or all j < m.
t
Consider a free resolution (2.6) for the module M and let Q be the matrix Pm−1
. Then
the following strong duality holds:
P
m
HK
(Rn , E 0 ) ' (H 0 (K, E Q ))0 = E Q [K].
(2.14)
Example 2.4. In the case of the Cauchy-Riemann operator, identifying R2n with
Cn , it is immediate to show that theorem 2.5 holds with m = n and, modulo a
sign, Q = P t . Using Poincare’s lemma, we then obtain that the dual of the sheaf of
holomorphic functions on a compact set, which is the space of hyperfunctions with
compact support, is isomorphic to the sheaf of holomorphic functions itself
30
H 0 (Cn , O) ' H n (Cn , O) ' (H 0 (Cn , O))0 .
(2.15)
This is the well known Köthe-Martineau-Sato duality theorem, which is at the core
of hyperfunction theory.
Since we are going to study this property of duality for other complexes, let us
give a definition.
Definition 2.1. We say that the operator Q(D) is s-dual to the operator P (D) if
Q
s
(Rn , E 0 ).
[H 0 (K, E P )]0 ∼
= HK
In particular, if Q = P t and the complex dual to the free resolution of the module
associated to P is isomorphic to the resolution itself, we say that the operator (or the
complex) is self-dual.
With this terminology, isomorphism (2.15) says that the Cauchy-Riemann operator in n complex variables is n-dual to itself.
2.3
The Cauchy-Fueter operator
The kernel of the Cauchy-Riemann operator on differentiable functions is the space
of holomorphic functions. Holomorphic functions are defined in the classical way as
those functions f : Ω ⊆ C −→ C such that the following limit exists at every point
z∈Ω
f (z + h) − f (z)
.
h→0
h
lim
31
If one wants to generalize this notion of holomorphicity to functions defined on the
space of quaternions
H := {q = x0 + i x1 + j x2 + k x3 | x0 , x1 , x2 , x3 ∈ R},
a possible natural approach would be to say that the limit in H
lim (f (q + h) − f (q)) · h−1
h→0
exists for every point q of the domain. This definition produces a much too narrow
class of functions. The only quaternionic functions for which the limits exists are the
linear functions
f (q) = aq + b,
a, b ∈ H.
A possible way out to this problem is to generalize the Cauchy-Riemann operator
and consider the nullsolutions of a first order differential operator for differentiable
functions on H ' R4 . The (left) Cauchy–Fueter operator is defined as follows:
∂q̄ := ∂x0 + i ∂x1 + j ∂x2 + k ∂x3 ,
(2.16)
so that we can introduce the notion of H-holomorphicity
Definition 2.2. A function f : Ω ⊆ H −→ H is regular if and only if satisfies
∂f
= 0,
∂ q̄
∀q ∈ Ω
Equivalently, f = f0 + i f1 + j f2 + k f3 is regular if its real components satisfy the
32
4 × 4 system of linear constant coefficients system of differential equations given by

f0
∂x0 −∂x1 −∂x2 −∂x3


 ∂x1
∂x2   f1
∂x0 −∂x3

 ∂x2
∂x3
∂x0 −∂x1   f2
∂x3 −∂x2
∂x1
∂x0
f3



 = 0.

(2.17)
Note that is it possible to define a right Cauchy-Fueter operator and the consequent notion of (right) regularity, but the theory that follows from this definition
would be identical. To simplify the notation, we will write the system (2.17) as a
matrix multiplication
U (D)f = 0.
When considering several quaternionic variables qi = xi0 + i xi1 + j xi2 + k xi3 , we will
write Ui (D)f = 0, so that system (2.17) for n Cauchy-Fueter operators becomes

U1 (D)


..


.

Um (D)


f0
f1 
 = 0,
f2 
f3


∂xi0 −∂xi1 −∂xi2 −∂xi3
 ∂xi1
∂xi0 −∂xi3
∂xi2 
.
where Ui (D) = 
 ∂xi2
∂xi3
∂xi0 −∂xi1 
∂xi3 −∂xi2
∂xi1
∂xi0
In several papers (see [2–4]) the authors have studied the information contained in
the module M = coker(P ) where P has entries in the ring of polynomials
R = C[x10 , x11 , x12 , x13 , . . . , xn0 , xn1 , xn2 , xn3 ]
and is given by the 4n × 4 matrix whose blocks are the Ui . The non commutativity
of the quaternionic setting makes non trivial the construction of the minimal free
resolution of M . A finite resolution of the module M can be calculated using CoCoA,
at least for some values of n.
33
Example 2.5. We can use the function Coala.FueterMat to write the symbol matrix
associated to any number Cauchy-Fueter operators, and Res to get the minimal free
resolution of M . Here is what we get in the case n = 1:
Use R::=Q[x[0..3]];
P:=Coala.FueterMat(Indets());
M:=Module(P);
Res(R^4/M);
0 --> R^4(-1) --> R^4
------------------------------From this we conclude that the Cauchy-Fueter system for one variable has no compatibility condition (in fact it is not overdetermined) since the only map in the resolution
is P t itself:
Pt
0 −→ R4 (−1) −→ R4 −→ 0.
It is also immediate to deduce that the first Ext module is nonzero so the Hartogs
phenomenon described in the previous section does not hold for regular functions in
one variable, exactly like in the case of holomorphic functions of just one complex
variable. CoCoA provides this result easily either with the use of the Ext package or
by simply checking that the matrix P has a nonzero determinant:
IsZero(Det(P));
FALSE
-----------------------------Example 2.6. In the case of 2 quaternionic variables the machine produces the
following output:
34
Use R::=Q[x[0..3]y[0..3]];
P:=Coala.FueterMat(Indets());
M:=Module(P);
Res(R^4/M);
0 --> R^4(-4) --> R^8(-3) --> R^8(-1) --> R^4
------------------------------Although the Betti numbers show a Koszul-like pattern, the degree 2 of the first
syzygy map tell us that this complex is not obtained by calculating the Koszul syzygies
of the pair (∂q¯1 , ∂q¯2 ). In fact, the two operators do not commute so we cannot expect
a Koszul-like behavior. Nonetheless, it is still possible to express the maps of the
resolution in terms of the Cauchy-Fueter operator and its conjugate
∂q := ∂x0 − i ∂x1 − j ∂x2 − k ∂x3 ,
(2.18)
If we denote by U1 , U2 the two by two blocks of P , by V1 , V2 the symbols of the
conjugate operators ∂q1 and ∂q2 , and by Di := Ui Vi the symbol of the Laplacian
operator
∆i = ∂q¯i ∂qi = ∂qi ∂q¯i = ∂x20 + ∂x21 + ∂x22 + ∂x23
then the Cauchy-Fueter complex is

h
i
V1 V2
0 −→ R4 (−4) −→ R8 (−3)

−D2 U1 V2
U2 V1 −D1
−→



U1 


U2
R8 (−1) −→ R4 −→ 0.
(2.19)
The resolution in this case shows that the compatibility conditions arise from the
fact that the Cauchy-Fueter operator and its conjugate commute with the Laplacian.
Again the first syzygies are quadratic but after that the complex is linear. The
35
elimination of compact singularities is possible because the first Ext module vanishes.
CoCoA confirms this by showing that the maximal minors of the matrix P are coprime:
GCD(Minors(1,P)); 1
------------------------------Probably the first interesting case is when n = 3. The compatibility conditions
for regular functions of three quaternionic variables have been calculated in [2]. They
unveiled an unexpected nature which greatly surprised the authors. That is, some
of the blocks appearing in the matrices of the resolutions could only be interpreted
in terms of operators which are not directly related to the Cauchy-Fueter operator.
Here is the resolution we obtain with CoCoA for the case n = 3
Pt
Pt
Pt
Pt
Pt
1
2
3
4
R4·3 (−1) −→ R4 −→ 0.
R4·10 (−3) −→
R4·15 (−4) −→
R4·9 (−5) −→
0 −→ R4·2 (−6) −→
(2.20)
At present, only the description of the matrices P1 and P4 in terms of quaternionic
blocks is explicit (see Chapter 5 for a more precise statement on the nature of the
syzygies in this complex). Let us begin with the first syzygies. The following theorem
provides a beautiful and compact description of all the compatibility conditions for
the Cauchy-Fueter system in n variables, hence including the case n = 3 as a corollary
(in fact, it suffices to describe the syzygies for the case of three operators and then
observe that those only involve 3 different indices at a time to conclude that they
hold for the general case).
36
Theorem 2.6. The compatibility conditions of the system

 ∂q̄1 f = g1
...
...

∂q̄n f = gn
with n > 2, are the following:
¡ ¢
(1) for each of the 2 n2 ordered pairs of indices r, s, 1 ≤ r, s ≤ n
∂q̄r ∂qs gs − ∂q̄s ∂qs gr = 0
(2) for each of the
¡n¢
3
triples of indices h, r, s, 1 ≤ h, r, s ≤ n
∂qh ∂q̄r gs + ∂qr ∂q̄h gs − ∂q̄s ∂qr gh − ∂q̄s ∂qh gr = 0
and
∂qr ∂q̄s gh + ∂qs ∂q̄r gh − ∂q̄h ∂qr gs − ∂q̄h ∂qs gr = 0,
(3) for each of the
¡n¢
3
triples of indices h, r, s, 1 ≤ h, r, s ≤ n
(Dqr ∂q̄s − Dqs ∂q̄r )gh + (Dqs ∂q̄h − Dqh ∂q̄s )gr + (Dqh ∂q̄r − Dqr ∂q̄h )gs = 0,
(Dq0 r ∂q̄s − Dq0 s ∂q̄r )gh + (Dq0 s ∂q̄h − Dq0 h ∂q̄s )gr + (Dq0 h ∂q̄r − Dq0 r ∂q̄h )gs = 0,
where
Dqi = −j ∂xi2 + k ∂xi3 ,
Dq0 i = −i ∂xi1 + k ∂xi3 .
Note how the syzygies of type (3) do not involve the Cauchy-Fueter operator or its
conjugate but only the operators D and D0 which involve two real variables instead
37
of four. Sometimes we call such operators ”Cauchy-Riemann like” operators because
of this. The quaternionic syzygies of type (1) and (2), which are written solely in
terms of ∂q and ∂q̄ will be referred to as radial syzygies (see the next section for a
motivation of this term), while the remaining syzygies (3) will be called exceptional.
Other properties of the Cauchy-Fueter complex, like the linearity of the resolution
after the first quadratic map and the exactness of the dual complex except for the
last spot, hold true like for the case n = 2. We will include all these facts, together
with the computation of the Hilbert function and the Betti numbers, in the following
theorem, with a sketch of the proof. For the individual statements and detailed proofs
see [3, 4].
Theorem 2.7. Let M be the module associated to the Cauchy-Fueter system in n > 1
variables. Then its resolution is
t
P2n−2
0 −→ Rr2n−1 (−2n) −→ Rr2n−2 (−2n + 1) −→ . . .
Pt
Pt
1
−→ Rr3 (−4) −→ Rr2 (−3) −→
R4n (−1) −→ R4 −→ M −→ 0.
In particular
I) the resolution of M has length 2n − 1;
II) all the maps in the resolutions are linear except for the first one.
III) the Betti number r` at the step ` is given by
µ
¶
2n − 1 n(` − 1)
r` = 4
.
`
`+1
38
Sketch of the proof. The proof of the theorem is exquisitely algebraic and makes use
of Gröbner Basis techniques, once again confirming a tight bound between this theory
and algebraic analysis of operators. First of all one can compute the reduced Gröbner
basis of the R-module hP t i that contains the columns of P t and the columns of the
matrices [Ut , Us ], where Ui is the symbol of the i-th operator and the pairing [ , ] is
the commutator. Next one can find define the module
M ∗ = R4 /hUt , Ur Us − Us Ur , x11 e` , xn2 e` , xi3 e` ii=1...n,
1≤r<s≤n, `=1,2,3,4
where ei is the canonical basis of R4 . Let ℘ be the maximal ideal of R generated by
all the 4n variables. Then, the variables x11 , xn2 , xi3 together with the polynomials
x21 + x12 , . . . , xi1 + xi−1,2 , . . . xn1 + xn−1,2 form a maximal M ∗ regular sequence in ℘.
Now using the Auslander-Buchsbaum formula one obtains that the module M has
projective dimension equal to 2n − 1 so the length of a minimal free resolution of M
is 2n − 1.
The fact that the first syzygies are quadratic follows by their explicit description
(see [3]), while the proof of the case of higher order syzygies follows by the computation
of the Castelnuovo-Mumford regularity of M .
Finally the Betti numbers can be computed by equating the coefficients of the
Hilbert function written using the minimal free resolution of M and the fact that the
Hilbert function of M is given by (see [3])
Pn (t) = 4
1 + (n − 1)t
.
(1 − t)2n+1
¤
The dual of the resolution arising from the Hilbert syzygy theorem is a complex that,
39
in general, is not exact. In this particular case the complex is
P
P
P2n−2
1
Rr2 −→ . . . −→ Rr2n−2 −→ Rr2n−1 −→ 0.
0 −→ R4 −→ R4n −→
(2.21)
By a well known result (see [60] Corollary 1, p. 337) we have immediately the
following:
Theorem 2.8. The Cauchy–Fueter complex (2.21) is exact except at the last spot,
i.e. Extj (M, R) = 0 for j = 1, . . . , 2n − 2, Ext2n−1 (M, R) 6= 0.
The proof of 2.8 is based on the calculation of the codimension of the characteristic
variety of the module M associated to the Cauchy-Fueter operator, which is the set
of all points in Hn ' R4n such that the rank of the matrix P is not maximal (or in
other words, the algebraic set in R4n where all the maximal minors of P vanish). The
codimension of the characteristic variety corresponds indeed with the index of the
last trivial Ext module. Since the dimension of the variety is 2n + 2, as calculated in
[4], we get the result. This method is yet another alternative for the characterization
of the vanishing of the cohomology modules. Needless to say, it can also be checked
with CoCoA, though again the representation of the variety as the ideal of minors has
the drawback of being rather expensive. One would enter the commands
V:=Ideal(Minors(4,P)); Dim(R/V);
and get the answer desired.
Let us conclude this paragraph with some analysis of the duality of the CauchyFueter complex.
Example 2.7. Let us consider P (D) = [∂q̄1 , ∂q̄2 ] acting on function f : H2 → H.
Then the results of [2] show that P (D) is 3-dual to Q(D)t = [∂q1 , ∂q2 ]. This duality
40
allows us to construct a theory of hyperfunctions as cohomology classes of quaternionic
regular functions in two variables. For more details on this theory of hyperfunctions,
see [25] and the results in [23, 24].
Remark 2.2. One could try to generalize the previous example to the case of three
quaternionic variables, by considering functions f : H3 → H, and by applying to such
functions three Cauchy–Fueter operators on three different variables. The computations carried out in [4] and in [25] show, however, that the duality is more complex
in this case and, in many senses, less satisfactory. Indeed, one can show the existence
of a matrix Q composed of nine 4 × 4 blocks per row, such that the symbol of the
Cauchy–Fueter operator in three quaternionic variables is 4-dual to Q. An explicit
expression of the operator Q(D) in n quaternionic variables will be given later on in
Chapter 5. We include here the case n = 3 for the sake of simplicity:
·
Q(D) =
∂q1 ∂q2 ∂q3
0
0
0
0
Ďq1
0
Ďq2
0
0
0
0
Ďq3 −Dq1 −Dq2 −Dq3
¸
(2.22)
where Ďqi = (∂x0i + i ∂x1i ).
To conclude this section, let us introduce a rather natural variation of the Cauchy–
Fueter operator: the Moisil–Theodorescu operator
∂q̃ = i ∂x1 + j ∂x2 + k ∂x3 .
As it is well known, the famous Frobenius theorem says that in R3 there are no
algebraic structures as good as those of C and H. This caused the idea to consider
functions of 3 real variables but ranged in H which led G. Moisil and N. Theodorescu
41
to consider the system

∂x3
f0
∂x2
0
∂x1


 ∂x1
0 −∂x3
∂x2   f1

 ∂x2
∂x3
0 −∂x1   f2
∂x3 −∂x2
∂x1
0
f3



=0

(2.23)
where the functions fj are real valued functions of three real variables. The system
(2.23) describes the kernel of the Moisil-Theodorescu operator. In [61] a full fledged
algebraic analysis of this operator have been developed. For the sake of brevity, we will
not report those results here. Let us point out that the analysis of this operator reflects
all the properties of the Cauchy-Fueter one. All the syzygies, the free resolution and
the Ext modules for the Moisil-Theodorescu operator can be obtained from the ones
of the Cauchy-Fueter operator ”by simply setting” x0 = 0. The function theory for
the space of solutions of (2.23) is then formally identical to the theory of regular
functions on H. In particular it has been shown that a duality theorem can be proved
for any dimension. The case of two variables (each of which is a vector in R3 ) is
the most immediate, and it can be proved that the Moisil–Theodorescu operator in
two variables is self-dual with s = 3 in the sense of Definition 2.1. The case of
three variables in R3 , however, gives rise to the same exceptional behavior of the
Cauchy-Fueter complex in three quaternionic variables.
2.4
The Dirac operator
The algebra of quaternions is just a special case of a more general class: the Clifford algebras. A real (resp. complex) Clifford algebra Cn is the associative algebra
42
generated by the n basis elements of Rn (resp. Cn ) with the following presentation:
Cn = he1 , . . . en | ei ej + ej ei = −2δij , i, j = 1 . . . ni
It is clearly a non commutative algebra. Its real (resp. complex) dimension is 2n , as
the set of elements
{e0 = 1} ∪ {eA = ea1 · · · eai | 1 ≤ i ≤ n, 1 ≤ a1 < · · · < ai ≤ n}
forms a basis for Cn called the basis of multivectors. The length of a multivector eA
is the length |A| = i of the multiindex A in the (minimal) representation ea1 · · · eai
as above. We set the length of 1 to be zero. Note that the Clifford algebra C2 is
isomorphic, as a real algebra, to the field of quaternions H with the identification
e0 = 1,
e1 = i ,
e2 = j ,
e12 = i j = k .
More in general, it is possible to show that every Clifford algebra is an algebra of
matrices with entries in one of the rings R, C, H, R ⊕ R or H ⊕ H. For a complete
classification see [25]. Let us now introduce a notion of regularity for real functions
with values in Cn . Indicating with x = (x1 , . . . , xn ) the real variables, we define the
Dirac operator acting on functions f : Rn −→ Cn as the linear operator
∂x =
n
X
ei ∂xi .
(2.24)
i=1
Definition 2.3. Let Ω ⊆ Rn be an open set and let f be a differentiable function
defined on Ω with values in the Clifford algebra Cn . We say that f is monogenic on
43
Ω if
∂x f (x) = 0,
for all x ∈ Ω.
Definition 2.3 is a generalization of the notion of holomorphicity on C and regularity on H as introduced in the previous sections. It is obviously possible to extend
this definition to the case of several copies of Rn . Naming xj = (xj1 , . . . , xjn ) the
real variables of the j-th copy of Rn in the cartesian product (Rn )k , we introduce the
Dirac operator
∂xj =
n
X
ei ∂xji .
(2.25)
i=1
and say that a function f : (Rn )k −→ Cn is monogenic if it is in the kernel of the
operator ∂xj for all j = 1 . . . k.
Problem 2.1. If one wants to carry out the algebraic analysis of the Dirac operator
(2.25), a representation of the operator as a real matrix is necessary. Looking at the
function f as a real valued function
n
f : (Rn )k −→ R2
n
via the identification Cn ' R2 , the problem is to find a way to represent the symbol
matrix P associated to a Dirac operator.
This has been addressed with CoCoA using the following schema: if we write f as
f=
X
|A|≤n
fA eA
44
then f is monogenic if and only if
n X
X
∂
∂xj f =
fA ei eA = 0.
∂xji
i=1
|A|≤n
The previous equation provides the entries of the 2n by 2n square matrix P . If we
identify each column of P with a multiindex B and each row with a multiindex A,
the entries of the matrix are precisely

 (−1)N ∂xji if ei eA = eB , N = |A| + |B| + i + 1
P (D)AB =

0
otherwise.
The implicit condition ei eA = eB can be rewritten in different ways using the algebra
of multivectors. It is immediate to check that the diagonal elements PAA are zero, like
most of the other entries of the matrix. Given two multi indices A and B, there exists
at most one basis vector ei such that ei eA = eB , and if we fix one of the multiindices,
say A, only n values of B are such that the entry PAB is non zero, and they correspond
to the n possible choices of ei . From this argument we can conclude that
Proposition 2.2. The symbol matrix P representing the Clifford operator ∂xj with
respect to the basis of multivectors of Cn is a sparse square matrix of dimension 2n in
which each row contains the variable xji , for each value of i between 1 and n, exactly
once.
Note that the same property described in the previous proposition holds for the
symbols of the Cauchy-Riemann and Cauchy-Fueter operators, although the matrices
are not sparse.
The functions that I have written for CoCoA to perform the calculation of the
45
symbol of the Dirac operator use a representation of the multivectors eA as the list of
indices [a1 , . . . , ai ]. Operations on lists such as juxtaposition and sorting mimic the
properties of product in Cn , though not very efficiently. With the use of the function
Coala.CliffMult it is then possible to write explicitly the matrix for any number k
of Dirac operators in, virtually, any dimension n. This is probably one of the first
examples in which CoCoA has been utilized for calculations in a non commutative
setting. Here are some examples of application.
Example 2.8. In the case of 2 Clifford operators on R3 , with variables x = (x1 , x2 , x3 )
and y = (y1 , y2 , y3 ) we can write the 16 by 8 matrix P with the command CliffordMat:
Use R::=Q[x[1..3],y[1..3]];
Coala.CliffordMat(Indets(),2);
Mat([
[0, x[1], x[2], x[3], 0, 0, 0, 0],
[-x[1], 0, 0, 0, -x[2], -x[3], 0, 0],
[-x[2], 0, 0, 0, x[1], 0, -x[3], 0],
[-x[3], 0, 0, 0, 0, x[1], x[2], 0],
[0, x[2], -x[1], 0, 0, 0, 0, x[3]],
[0, x[3], 0, -x[1], 0, 0, 0, -x[2]],
[0, 0, x[3], -x[2], 0, 0, 0, x[1]],
[0, 0, 0, 0, -x[3], x[2], -x[1], 0],
[0, y[1], y[2], y[3], 0, 0, 0, 0],
[-y[1], 0, 0, 0, -y[2], -y[3], 0, 0],
[-y[2], 0, 0, 0, y[1], 0, -y[3], 0],
46
[-y[3], 0, 0, 0, 0, y[1], y[2], 0],
[0, y[2], -y[1], 0, 0, 0, 0, y[3]],
[0, y[3], 0, -y[1], 0, 0, 0, -y[2]],
[0, 0, y[3], -y[2], 0, 0, 0, y[1]],
[0, 0, 0, 0, -y[3], y[2], -y[1], 0]
])
-------------------------------
and we can also transform this CoCoA matrix into a definition of a matrix for Singular
with the command Coala.MatToSingular.
Using the usual algebraic approach described in section 2.2 we can study some
analytical problems associated to a system of equations

 ∂x1 f (x1 , . . . , xk ) = 0

∂xk f (x1 , . . . , xk ) = 0
In particular, we are able to find the compatibility conditions of the associated inhomogeneous system

 ∂x1 f (x1 , . . . , xk ) = g1

(2.26)
∂xk f (x1 , . . . , xk ) = gk
and the free resolution of the associated module. This has been done in the case
of two and three Dirac operators in the paper [63] where all the syzygies of the
system are presented explicitly, at least in the case of n ”large enough”. It has been
conjectured and proved using a particular polynomial decomposition (the so called
47
Fisher Decomposition) that if n ≥ 2k − 1, the relation satisfied by k Dirac derivatives
are of the following form:
[{∂x i , ∂x j }, ∂x k ] = 0.
(2.27)
Such relations are called radial relations. When the real dimension of Cn is smaller, the
radial relations are still satisfied, but in addition to them, some extra relations hold
for the complex. This is the case, for example, of the exceptional syzygies described in
Theorem 2.6 for the Cauchy-Fueter operator. To introduce the first non-exceptional
case, I will present here the results from [63] for the case of 3 Dirac operators in real
dimension n ≥ 5.
Remark 2.3. Since the first syzygies involve only three operators at a time, the
description of the map P1 is actually a general description of the first compatibility
conditions for any number of operators.
CoCoA produces the following resolution for three operators on R5 . The Betti
numbers have been factored by 2n so that they show the actual number of Dirac
relations in the complex.
n
n
n
n
n
0 −→ R2 (−9) −→ R3·2 (−8) −→ R8·2 (−6) −→ R6·2 (−4) ⊕ R6·2 (−5) −→
n
n
n
−→ R8·2 (−3) −→ R3·2 (−1) −→ R2 −→ M3 −→ 0.
This means that we have a total of
- 3 generators for the initial module, corresponding to the 3 Dirac operators
- 8 quadratic syzygies at the first step
- 6 linear and six quadratic second syzygies
(2.28)
48
- 8 quadratic syzygies at the third step
- 3 linear syzygies at the second to last step
- one linear syzygy that closes the complex
What this description does not show is the explicit form of the syzygies. They have
been calculated ”by hand” and with the help of CoCoA in [63]. Here I summarize
such result, whose notation I will use throughout the dissertation and in particular
in Chapter 6.
Theorem 2.9. Let f, g1 , g2 , g3 : (Rn )3 → Rn , n ≥ 5 be differentiable functions and
let ∂xj , j = 1, 2, 3 be the Dirac operator on each copy of Rn . Consider the Dirac
system

 ∂x1 f = g1
∂x f = g2
 2
∂x3 f = g3
(2.29)
where f, g1 , g2 , g3 : (Rm )3 → Rm , m ≥ 5.
First Syzygies. The first compatibility conditions on the gi are given by





























∂x2 ∂x1 g1 − ∂x21 g2 = h12
∂x3 ∂x1 g1 − ∂x21 g3 = h13
∂x1 ∂x2 g2 − ∂x22 g1 = h21
∂x3 ∂x2 g2 − ∂x22 g3 = h23
∂x1 ∂x3 g3 − ∂x23 g1 = h31
∂x2 ∂x3 g3 − ∂x23 g2 = h32
{∂x2 , ∂x3 }g1 − ∂x1 ∂x2 g3 − ∂x1 ∂x3 g2 = a1
{∂x1 , ∂x3 }g2 − ∂x2 ∂x1 g3 − ∂x2 ∂x3 g1 = a2
{∂x1 , ∂x2 }g3 − ∂x3 ∂x1 g2 − ∂x3 ∂x2 g1 = a3
(2.30)
with the constraint a1 + a2 + a3 = 0. This makes 8 independent relations.
Second Syzygies. The 6 linear compatibility conditions of system (2.30), that give
49
the syzygies at the second step, are given by















∂x2 h12 + ∂x1 h21
∂x3 h13 + ∂x1 h31
∂x3 h23 + ∂x2 h32
∂x3 h12 + ∂x2 h13
∂x1 h23 + ∂x3 h21
∂x2 h31 + ∂x1 h32
=0
=0
=0
= ∂x1 a1
= ∂x2 a2
= ∂x3 a3
while the 6 quadratic compatibility conditions are given by the permutations of (1, 2, 3)
in
{∂x1 , ∂x2 }h23 + ∂x22 a3 = ∂x3 ∂x2 h21 ,
of which only 3 are independent. Finally, we have the permutations of the condition
∂x21 h23 − ∂x22 h13 = ∂x3 ∂x1 h21
which gives 3 (independent) more quadratic relations. Let us write explicitly a system
of 6 independent relations for the second syzygies as follows:

∂x3 h23 + ∂x2 h32 = R1 ,




∂x h31 + ∂x3 h13 = R2 ,


 1
∂x1 h21 + ∂x2 h12 = R3 ,
 ∂x3 h12 + ∂x2 h13 − ∂x1 a1 = S1



∂ h + ∂x3 h21 − ∂x2 a2 = S2


 x1 23
∂x2 h31 + ∂x1 h23 − ∂x3 a3 = S3
then the 6 equations given by the permutations of (i, j, k) = (1, 2, 3) in
{∂xi , ∂xj }hjk + ∂x2j ak − ∂xk ∂xj hji = Tki
(2.31)
50
and, finally, by the 6 permutations of (i, j, k) = (1, 2, 3) in
∂xi ∂xj hkj + ∂x2k hji − ∂x2j hki = Ukj .
This would lead to a total of 18 relations, but in addition we have the following
constraints on Tij and Uij :
T23 + T32 = ∂x1 S1 ,
T31 + T13 = ∂x2 S2 ,
T12 + T21 = ∂x3 S3
U23 + U32 = ∂x1 R1 ,
U31 + U13 = ∂x2 R2 ,
U12 + U21 = ∂x3 R3 .
These constraints reduce the total number of equations in the system to 12. Basically,
all the linear relations are necessary at this step, while for the quadratic we can just
keep one Tij and one Uij for each ordered pair of indices (i, j) such that i < j.
Third Syzygies. The 8 compatibility conditions of the previous system are given by
1. (a) the 3 conditions obtained by the cyclic permutations of (1, 2, 3) in the formula
∂x2 U13 − ∂x1 U32 + ∂x22 R2 − ∂x23 R3 = 0
of which 2 are independent;
2. (b) the 6 permutations of the relation
∂x21 S2 + ∂x2 T23 − ∂x3 ∂x1 R3 − ∂x1 U12 = 0.
We have now the non–homogeneous system consisting of
∂x21 S2 + ∂x2 T23 − ∂x3 ∂x1 R3 − ∂x1 U12 = B12
∂x22 S1 + ∂x1 T13 − ∂x3 ∂x2 R3 − ∂x2 U21 = B21
51
∂x21 S3 + ∂x3 T32 − ∂x2 ∂x1 R2 − ∂x1 U13 = B13
∂x23 S1 + ∂x1 T12 − ∂x2 ∂x3 R2 − ∂x3 U31 = B31
∂x22 S3 + ∂x3 T31 − ∂x1 ∂x2 R1 − ∂x2 U23 = B23
∂x23 S2 + ∂x2 T21 − ∂x1 ∂x3 R1 − ∂x3 U32 = B32
∂x1 U32 − ∂x3 U21 + ∂x22 R2 − ∂x21 R1 = C1 ,
∂x2 U13 − ∂x1 U32 + ∂x23 R3 − ∂x22 R2 = C2 ,
∂x3 U21 − ∂x2 U13 + ∂x21 R1 − ∂x23 R3 = C3 ,
(2.32)
with the constraint C1 + C2 + C3 = 0.
Fourth Syzygies. For this system, we have that the 3 compatibility conditions are
given by
{∂x1 , ∂x2 }C3 + ∂x2 ∂x1 C2 = ∂x3 ∂x2 B12 + ∂x22 B13 − ∂x3 ∂x1 B21 − ∂x21 B23
and its cyclic permutations, which give the system of three equations
E1 = {∂x2 , ∂x3 }C1 + ∂x3 ∂x2 C3 − ∂x1 ∂x3 B23 − ∂x23 B21 + ∂x1 ∂x2 B32 + ∂x22 B31
E2 = {∂x3 , ∂x1 }C2 + ∂x1 ∂x3 C1 − ∂x2 ∂x1 B31 − ∂x21 B32 + ∂x2 ∂x3 B13 + ∂x23 B12
E3 = {∂x1 , ∂x2 }C3 + ∂x2 ∂x1 C2 − ∂x3 ∂x2 B12 − ∂x22 B13 + ∂x3 ∂x1 B21 + ∂x21 B23
(2.33)
Last Syzygy. Using C1 + C2 + C3 = 0, the last relation that closes the complex is
∂x1 E1 + ∂x2 E2 + ∂x3 E3 = 0,
(2.34)
52
which closes the complex.
Other experiments have been performed using Singular on the cluster [20], trying
to get a pattern in the Betti numbers for the various values of k and n. The following
charts display the Betti numbers obtained with the calculations I ran on the machine
for the case of 3 and 4 Dirac operators. Note that all numbers have been divided by
a factor of 2n in order to show the actual number of Dirac relations. Some interesting
k=3
C1
C2
C3
C4
C5
C6
C7
C8
C9
r0
1
1
1
1
1
1
1
1
1
r1
3
3
3
3
3
3
3
3
3
r2
3
3
10
10
8
8
8
8
8
r3
r4
1
1
15
9
15
9
12
8
12
8
12
8
12
8
(12) (8)
r5
r6
r7
2
2
3
1
3
1
3
1
3
1
(3) (1)
Table 2.1: Betti numbers for three Dirac operators
facts can be observed. First, the resolution for an even dimension 2m is the same as
the one obtained for 2m − 1. This fact can be proven in general using a particular
decomposition of the Clifford algebra and the invariance of the Dirac operator with
respect to the space SpinR (m) [25]. Therefore, we one can limit the analysis to n odd.
Second, while n = 1 and n = 2 produce a Koszul-type of sequence of Betti numbers,
the cases n = 3, 4 present exceptional relations exactly like the case of 3 Cauchy-Fueter
operators. This is due to the fact that it is possible to reduce the Dirac operator on
C4 to the Cauchy-Fueter operator again using Spin(4)-invariance. Finally for n > 4
the complex becomes ”purely radial” and follows the results of theorem 2.9. Note
53
that the number in parenthesis, for n = 9, have not been confirmed experimentally
but are a consequence of such theorem. In the case of 4 operators, we do not have a
complete picture as the following chart shows:
k=4
C1
C2
C3
C4
C5
C6
C7
C8
r0
1
1
1
1
1
1
1
1
r1
4
4
4
4
4
4
4
4
r2
6
6
28
28
25
25
20
20
r3
4
4
70
70
92
92
56
56
r4
1
1
84
84
168
168
> 64
> 64
r5
r6
56 20
56 20
165 140
165 140
... ...
... ...
r7
r8
3
3
...
...
...
...
...
...
...
...
Table 2.2: Betti numbers for four Dirac operators
The values . . . correspond to ranks that were not possible to calculate (in a reasonable time within the possibilities of [20]). The values > 64 indicate that a calculation
of the Gröbner Basis for the syzygy module was started and by giving the command
Set Verbose; on CoCoA we were able to monitor partial calculations and find out
that there are at least 64 syzygies. This gives us an idea of the complexity of the
computation of syzygies for this case. Not only the dimension of Cn contributes exponentially to the number of generators, but we also have to keep in mind that based on
a famous paper of D. Bayer and M. Stillman [7], the intrinsic complexity of the computation of syzygies is doubly exponential in the number of generators of the module.
Anyhow, the Betti numbers for the cases n = 1, 2 still follow a Koszul-type pattern
like in the case of three operators. For 3 ≤ n ≤ 6 we have exceptional syzygies, but
we do not know the exact length of the complex unless for the ”quaternionic” case
n = 4. Starting from n = 7 we begin to see numbers corresponding only to the radial
54
syzygies, but we cannot reach the end of the complex either. The algebraic analysis of
the Dirac operator is then yet to be investigated. I hope that the growth of the power
of the computers we are able to use will enable us to dig deeper into the analysis and
perform further experiments. In Chapter 6 I will present an alternative approach to
using the real representation for the radial case.
Chapter 3: Surjectivity theorems and regularity of
differential operators
One of the early successes of the theory of locally convex topological vector spaces was
the proof that linear constant coefficients differential operators act surjectively on a
large class of functional spaces. For example, if P is a polynomial in C[z1 , . . . , zn ], D
is the differential operator (−i∂x1 , . . . , −i∂xn ) and O(Ω) is the space of holomorphic
functions on a convex open set Ω in Cn , then one can show that P (D) is surjective
from O(Ω) to itself. This particular result is a direct consequence of the fact that
the dual space of O(Ω) can be identified with a space of entire functions satisfying
suitable growth conditions, and of the classical Lindelöf theorem.
As I introduced in section 2.2, in the recent years there has been a rebirth of interest
in the theory of regular functions in one or more quaternionic variables. In this case,
however, the tools which allow the immediate proof of the surjectivity of P (D) on
O(Ω) are not available. The main purpose of this chapter is to show how to use
some of the techniques from the algebraic analysis of constant coefficients differential
operators to obtain a surjectivity result on the space of regular functions. Specifically,
we will prove a general result which has as a consequence the following theorems (see
Section 3.3 for notation):
Theorem 3.1. For any polynomial p(q) with complex coefficients, the operator p(∂q )
is surjective on the space of regular functions.
55
56
Theorem 3.2. Any operator of the form
∂m
m
m
m
∂x11 ∂x22 ∂x33
is surjective on the space of
regular functions on convex open sets in H.
The starting point is to note that the surjectivity of a differential operator on
the space of holomorphic functions is a consequence of the triviality of its syzygies.
When we consider operators on regular functions, however, this triviality does not
hold anymore, since even the Cauchy–Fueter operator is in fact a system of operators.
I will show how we can compute the appropriate syzygies so that, even though they
may not be trivial, we can still use their explicit form to deduce the surjectivity of
the operators we are interested in.
This chapter collects all the results from our paper [22] and presents some more
examples of applications. It is divided as follows. In Section 3.1 I recall the basic
construction for syzygies of systems of constant coefficients differential operators.
Section 3.2 is devoted to the proof of an algebraic result (of independent interest) on
the Koszul complex associated to systems of differential operators. Finally Section
3.3 provides the consequences to the study of differential equations, and in particular
to the case of regular functions. Throughout the chapter I will also include some
examples in which CoCoA is used to verify the hypotheses for the applicability of
the theorems. No particular procedure has been coded for this scope, as most of the
functions needed in this case are already available with the standard release of CoCoA.
57
3.1
Systems of differential equations
Let us consider a system of linear constant coefficients partial differential equations
of the form
(p1 (D), . . . , pk (D))f = P (D)f = 0,
where p1 , . . . , pk are polynomials in R = C[z1 , . . . , zn ], D = (−i∂x1 , . . . , −i∂xn ) and
f : U ⊆ Rn → R is, as always, an infinitely differentiable function on an open convex
set. It has been discussed in Chapter 2 that some important information on the system
is contained in the polynomial matrix P1 whose rows generate the first syzygies of the
k-vector P t = [p1 , . . . , pk ]. For example we know that the system P (D)f = g has a
differentiable solution f if and only if g satisfies the system P1 (D)g = 0. When P (D)
is a single operator, then P1 (D) ≡ 0, and so the surjectivity of P (D) follows. More
generally, when the sequence of polynomials p1 , . . . , pk is regular (see [57]), the matrix
P1 can be constructed in a simple way because its rows consist of all the vectors of
the type
[0, . . . , − pj , 0, . . . , 0, pi , . . . , 0].
|{z}
|{z}
i − place
j − place
One can repeat this procedure to find the syzygies of P1 , and so on, until one obtains
the so called Koszul complex that is nothing but the minimal free resolution of the
module M = R/I, where I is the ideal generated by p1 , . . . , pk . When considering
more general systems, in which the unknown function f is a vector and the matrix
P (D) of differential operators acting on it is an r1 × r0 matrix (in the scalar case
r1 = 1), the situation becomes more complicated. The theory of Gröbner Bases
offers some algorithms to compute the first syzygy module, but the complexity of the
58
problem is in general doubly exponential in the number of variables as shown in the
paper [7]. Moreover, the procedure of computing syzygies with Gröbner Bases does
not take into account the structure of the matrix P : even though in some cases P can
be seen as a block matrix, in which every block represents an operator (e.g. the cases
presented in the previous chapter follows into this category) a priori the matrix of the
first syzygies is not necessarily written in terms of those blocks; this is dramatically
demonstrated in Theorem 2.6 for the case of the Cauchy–Fueter system in 3 or more
quaternionic variables. On the other hand, given a matrix


P1 (D)
P (D) =  . . . 
Pk (D)
where each Pi (D) is a n × n matrix, it might be interesting to understand under
which conditions it is possible to build a Koszul complex starting from the matrices
Pi . Obviously, it is not always possible to construct such a complex, mainly because
of the non commutativity of the ring of matrices and one may wonder what are the
conditions under which the procedure can be applied. Our results will allow us to
calculate the minimal free resolution rather easily, at least in some cases.
3.2
Regular sequences of matrices
In this section we study a system of differential equations of the form P1 (D)f = · · · =
Pk (D)f = 0 and we seek conditions on the polynomial square matrices P1 , . . . , Pk such
that the compatibility conditions on the system, i.e. the first syzygies of the rows of
59
the matrix


P1
 
P =  ... 
Pk
can be easily written in terms of the matrices Pi themselves. For example, given
k = 2, if the matrices P1 and P2 commute, we may expect that the syzygies are the
rows of the matrix [−P2 P1 ]. This is suggested by the fact that the following matrix
product is zero:
£
−P2
· ¸
¤ P1
P1
= 0.
P2
This commutativity, however, is not enough to guarantee that [−P2 P1 ] contains all
the syzygies we need as we can easily see for example taking P1 = P2 . In general,
the validity of this result depends on whether or not we have relations between rows
of P1 and rows of P2 , or among rows of the same matrix. According to well known
results in algebra, the situation can be fully understood through suitable algebraic
conditions on the matrices involving the notion of regular sequence. In the sequel,
we will denote the ring of n × m matrices with entries in the ring R by Matn,m (R),
while, when n = m, we will write Matn (R); if R is an integral domain, Frac(R) will
denote its field of fractions.
Definition 3.1. Let R be a ring, and let P1 , . . . , Pk be matrices in R = Matn (R).
We say that the k-uple (P1 , . . . , Pk ) is a left regular sequence if:
1) P1 is a left regular element of R, i.e. the only B ∈ R such that BP1 = 0 is B = 0;
2) Pi is not a zero divisor in R/(P1 , . . . , Pi−1 )R for all i = 2, . . . , k where (P1 , . . . , Pi−1 )R
is the left ideal in R generated by P1 , . . . , Pi−1 .
60
When k = 1 we have the square system P1 (D)f = 0 and the condition that P1 is
a non-zerodivisor in R is fully equivalent to the fact that we do not have nonzero
syzygies for its rows, as stated by the following proposition.
Proposition 3.1. Let R be an integral domain. The following are equivalent facts
for a square matrix P ∈ R:
1) Det(P ) 6= 0;
2) P is a left regular element of R;
3) Syz(P ) = h0i ⊂ Rn where Syz(P ) means the first module of the syzygies for the
rows of the matrix P .
Proof. 1)⇒ 2): let B be a nonzero square matrix such that BP = 0. Then, in
particular, any row of B is a solution to the linear system (x1 . . . xn )P = (0 . . . 0)
which has only trivial solutions since Det(P ) 6= 0. Hence B = 0.
2)⇒ 3): if x = (x1 . . . xn ) ∈ Syz(P ) is a nonzero row, than the matrix X whose n
rows are all equal to x is such that XP = 0 but this is a contradiction since P is left
regular.
3)⇒ 1): suppose Det(P ) = 0. Then the linear system xP = 0 has a non trivial
solution (x1 . . . xn ) ∈ Frac(R)n , and multiplying this solution by the product of all
nonzero denominators of its elements, we get a non trivial syzygy (s1 . . . sn ) ∈ Rn .
If we require the condition of regularity on the matrices of an overdetermined system,
then we are able to describe the module of the first syzygies. This fully corresponds to
what we would have obtained if we considered the syzygies of k polynomials g1 , . . . , gk
forming a regular sequence. The free resolution for the ideal I = (g1 , . . . , gk ) in this
61
case is the Koszul complex, and in particular the first syzygies are of the form
Syz(g1 , . . . , gk ) = h(0, . . . , −gj , 0, . . . , 0, gi , . . . , 0)| i < ji.
If we consider for example three commuting matrices P1 , P2 , P3 forming a left regular
sequence, then the syzygies can be written as the rows of the following matrix:


−P2 P1
0
−P3
0 P1  .
0 −P3 P2
The commutativity of the matrices Pi implies that these are syzygies, but in general
we could expect to find more relations among their rows. The following result, in
which R still denotes the ring Matn (R), assures that this is not the case.
Theorem 3.3. Let P1 , . . . , Pk ∈ R be k > 1 matrices such that:
1) Pi Pj = Pj Pi for all i, j = 1 . . . k,
2) (P1 , . . . , Pk ) is a left regular sequence,
and let M be the R-submodule of Rn generated by the rows of P1 , . . . , Pk . Then the
first syzygy module Syz(M) is generated by the rows of the block matrix



B=

B1
B2
..
.
Bk−1



,


0
0
Bi = 

0
. . . 0 −Pi+1 Pi 0
. . . 0 −Pi+2 0 Pi
...
...
. . . 0 −Pk 0 0
...
...
...
...

0
0
.

Pi
(3.1)
Proof. Let us define P to be the block matrix where we put the Pi ’s in a column.
Then it is easy to see that we have B · P = 0 since the Pi ’s commute, and that means
that the rows of the matrix B are syzygies for the rows of the matrix P , i.e. they
62
belong to the set of generators of M. Conversely, let us consider a nonzero row vector
x = (x11 , . . . , x1n , . . . , xk1 , . . . , xkn ) ∈ Syz(M).
This means that the product x · P is the zero vector of Rkn . We will show that x
belong to the R-module generated by the rows of B. Let us consider the matrix
X ∈ Matn,kn (R) containing n rows, all equal to x. Obviously X · P is the zero n × n
matrix. Then we can think of X as a block matrix of k square matrices:
£
X = X1 X2 . . .
Xk
¤
and so the product X · P = 0 means
X1 P1 + · · · + Xk Pk = 0.
(3.2)
We now proceed by induction on k.
Case k=2
Equation (3.2) for k = 2 is simply X1 P1 + X2 P2 = 0 that implies X2 P2 = −X1 P1 ,
which is possible only if the matrix X2 is a left multiple of P1 , given the fact that
(P1 , P2 ) is a regular sequence. So there exists a square matrix F ∈ R such that
X2 = F P1 and hence we get
0 = X1 P1 + F P1 P2 = X1 P1 + F P2 P1 = (X1 + F P2 )P1 ,
(3.3)
which implies, P1 being a left regular element, that X1 = −F P2 , i.e. the matrix X is
of the form
[X1 X2 ] = F · [−P2 P1 ] = F · B̃1
(3.4)
where B̃1 is the only block in the matrix B that we have in this case. In particular
63
(3.4) says that a row of X is a combination of the rows of B̃1 whose coefficients are
a row of F .
Case k>2
In this case we can rewrite equation (3.2) as
Xk Pk = −X1 P1 − · · · − Xk−1 Pk−1 .
(3.5)
Using the hypothesis of regularity, in particular the fact that Pk is regular in the quotient R/(P1 , . . . , Pk−1 )R, we know that Xk has to be in the left ideal (P1 , . . . , Pk−1 )R:
Xk = F1 P1 + · · · + Fk−1 Pk−1 .
(3.6)
Plugging this equation into (3.5) we have
F1 P1 Pk + · · · + Fk−1 Pk−1 Pk = −X1 P1 − · · · − Xk−1 Pk−1
(3.7)
that can be rewritten using the commutativity of the matrices as
(F1 Pk + X1 )P1 + · · · + (Fk−1 Pk + Xk−1 )Pk−1 = 0
(3.8)
which means that the rows of [(F1 Pk + X1 ) . . . (Fk−1 Pk + Xk−1 )] are syzygies for
[P1 . . . Pk−1 ]. The first k − 1 matrices of P obviously commute and are still a left
regular sequence, so we can apply the inductive hypothesis and get their syzygies as
combination of the rows of the matrix



B0 = 

B10
B20
..
.
0
Bk−2





,

0

0
Bi0 = 

0
...
...
...
...
0 −Pi+1 Pi 0
0 −Pi+2 0 Pi
...
0 −Pk−1 0 0
...
...
...
...

0
0


Pi
64
where the blocks Bi0 have the same rows and columns of the blocks Bi except for the
last n rows and the last n columns. Every matrix of the type [Y1 . . . Yk−1 ] whose
rows are syzygies for P1 , . . . , Pk−1 is obtained by a suitable combination of the rows
of B 0 , in particular there exists a set of square matrices
{Fji ∈ R| i = 1, . . . , k − 1, j = i + 1, . . . , k − 1}
such that we have Y1 =
Pk−1
Yl = −
j=2
l−1
X
Fj1 Pj , Y2 = −F21 P1 +
Fli Pi +
i=1
k−1
X
Fjl Pj ,
Pk−1
j=3
Fj2 Pj , and in general
l = 1, . . . , k − 1.
j=l+1
Hence, using the coefficients Fl Pk + Xl from (3.8) as Yl ’s and defining Fkl := −Fl for
l = 1, . . . , k −1, the equality (3.6) becomes Xk = −Fk1 P1 −· · ·−Fkk−1 Pk−1 . Moreover
we have the following:
Xl = −
l−1
X
i=1
Fli Pi +
k
X
Fjl Pj ,
l = 1, . . . , k,
j=l+1
which express the matrix X as a R-combination of elements of B, namely

B1


¤  B2 

Fkk−2 ) (Fkk−1 ) 
 ... .
Bk−2 
Bk−1

£
(F21 . . .
Fk1 ) (F32 . . . Fk2 ) . . .
(Fk−1k−2
To conclude the proof, it suffices to note that x is a row of X and so it is itself a
combination of rows of B.
Remark 3.1. Since the matrices Pi commute, and since they are a regular sequence
65
in R, it is obvious (and follows from the general theory) that the syzygies of the ideal
generated by P1 , . . . , Pk in R are given by the matrix B. What our result proves is
the much stronger fact that B actually gives the module of syzygies for the module
generated in Rn by the rows of P1 , . . . , Pk .
Remark 3.2. Note that our construction works also in the case in which R is any
other commutative ring, for example the ring Exp0 (Cn ) of entire functions of infraexponential type, which is the ring of the symbols of infinite order differential operators
(see [25]).
As anticipated before, we can now construct the whole free resolution for the
module M using the hypothesis of regularity.
Theorem 3.4. Let P1 , . . . , Pk ∈ R be k > 1 matrices such that:
1) Pi Pj = Pj Pi for all i, j = 1 . . . k,
2) (P1 , . . . , Pk ) is a left regular sequence,
and let M be the R-submodule of Rn generated by the rows of P1 , . . . , Pk . The resolution of M is formally constructed as the classical Koszul complex:
k
k
k
k
0 → Rn·(k) → Rn·(k−1) → . . . → Rn·(2) → Rn·(1) → 0.
If we consider the case of just two matrices P and Q, which is the situation we
are generally interested in, it is possible to rewrite condition 2) of Theorem 3.3 in an
equivalent more operative way as follows:
Proposition 3.2. Let R be an integral domain and set R = Matn (R). Let P, Q ∈ R
be square matrices such that P Q = QP . Then the following conditions are equivalent:
a) P and Q form a left regular sequence in R,
66
b)
1. Q is invertible in Matn (Frac(R)),
2. for every A ∈ R such that AQ−1 P ∈ R, we have AQ−1 ∈ R.
Proof. The fact that Q is invertible in Matn (Frac(R)) is equivalent to the request
that Det(Q) 6= 0 which is the same as Q being a regular element by Proposition 3.1.
Consider the relation X1 P + X2 Q = 0, which, using the hypotheses, can be written
as X1 Q−1 P = −X2 . This means that X1 Q−1 P belongs to R, so X1 Q−1 ∈ R from
which it follows that X1 = AQ for some A and X2 = −AP . If, conversely, (P, Q) is
left regular, it follows that for every A such that AP = BQ we have A = CQ, and so,
inverting the matrix Q, the second part of statement b) is true in Matn (Frac(R)).
Remark 3.3. Note that the ring R of the entries of the matrices P and Q could be
any integral domain. If P and Q, as we will consider in several examples of the next
section, are the matrices of symbols of a finite order linear differential operator, we
can take R as the ring of polynomials over C, but it is also possible to consider infinite
order differential operators, so that in this case we have holomorphic functions (in
fact infraexponential entire functions) as entries. In that case the second condition
in part b) of proposition 3.2 becomes:
AQ−1 P with infraexponential entries ⇒ AQ−1 with infraexponential entries.
3.3
Surjectivity results and some applications
In the case in which P1 and P2 satisfy the hypothesis of Theorem 3.3 or the equivalent
¸
·
P1
reformulation in Proposition 3.2 then one immediately has that the syzygies of
P2
are given by [−P2 P1 ]. We have the following corollaries of Theorem 3.3.
67
Corollary 3.1. Let P1 , P2 be two commuting matrices forming a regular sequence in
·
¸
P1 (D)
R = Matn (R). Then the range of the operator P (D) =
is given by
P2 (D)
{(g1 , g2 ) ∈ C ∞ | P2 (D)g1 = P1 (D)g2 }.
Corollary 3.2. Let P1 , P2 be two commuting matrices forming a regular sequence in
R. Let Q = Ker(P2 ) and let U be an open convex (or compact convex) set, then the
operator
P1 (D) : Q(U ) → Q(U )
is surjective.
More in general:
Corollary 3.3. Let P1 , . . . , Pk be commuting matrices forming a regular sequence in
R and let Q = {f | P2 (D)f = . . . = Pk (D)f = 0}. Let U be an open convex (or
compact convex) set, then the operator
P1 (D) : Q(U ) → Q(U )
is surjective.
A large class of examples to which this theory applies can be found in the quaternionic setting. Let us consider the pair of operators (∂q̄ , p(∂q )) where ∂q̄ is the CauchyFueter operator while p(∂q ) is a polynomial in ∂q with complex coefficients. Considering their Fourier transforms we obtain the pair (q̄, p(q)) in the ring R = C[q, q̄] which
forms a regular sequence since q̄ is not a factor of p(q). This fact can be usefully
reformulated in the framework of matrices.
68
Let us consider the 4 × 4 matrix Q representing the Fourier transforms of the
matrix Q(D), corresponding to the operator ∂/∂ q̄. Similarly, let P be the Fourier
transform of the matrix P (D) associated to p(∂q ). We have the following result:
Proposition 3.3. The pair (Q, P ) form a regular sequence in the ring R = Mat4 (R),
where R = C[x0 , . . . , x3 ].
Proof. To prove the statement we will use Proposition 3.2. Note that the matrix
associated to the operator ∂q is Qt , and QQt = (
the matrix Q−1 = (
P3
j=0
P3
j=0
x2j )I. So the inverse of Q is
x2j )−1 Qt . Moreover, since Q and Qt commute also P and Q
commute. Let now A be a matrix such that AQ−1 P ∈ R; the elements in the matrix
Q−1 all have denominator equal to
P3
j=0
x2j . The matrix P that does not contain Q
as a factor, so the only possibility to have AQ−1 P with polynomial entries is that A
contains the factor Q.
Combining the previous discussion with Proposition 3.3, we obtain that
Corollary 3.4. For any polynomial p(q) with complex coefficients, the operator p(∂q )
is surjective on the space of regular functions.
Remark 3.4. Note that an analogous result can be obtained for the pair of operators
P
∂
(∂x , p(∂x )) where ∂x = m
j=1 ej ∂xj is the Dirac operator on the Clifford algebra Cn
defined in section 2.4, and p(∂x ) =
PN
l=0
al ∂xl is a polynomial in ∂x with complex
¯ = −∂x and
coefficients. Note in fact that the conjugate of the Dirac operator is ∂x
¯ = −∆m .
¯
moreover ∂x∂x
= ∂x∂x
The previous approach can be also used to treat another interesting case (see also
[39], Theorem 4.2). Let us introduce the following notations: let ν = {λ1 , . . . , λm }
69
be a set of integers with λ1 , . . . , λm ∈ {1, 2, 3} and let ni be the number of λi ’s. We
will denote by σm the set of triples [n1 , n2 , n3 ] such that n1 + n2 + n3 = m. Let us
consider the pair (∂/∂ q̄, F (D)) where
F (D) =
N X
X
pν (D)aν ,
aν ∈ H
m=0 ν∈σm
and
X
1
pν (D) =
m! 1≤λ ,...,λ
1
m ≤3
where the sum is taken over the
µ
¶ µ
¶
∂
∂
∂
∂
−
iλ · · ·
−
iλ
,
∂xλ1
∂x0 1
∂xλm
∂x0 m
n!
n1 !n2 !n3 !
different alignments of ni elements equal to
i, with i = 1, 2, 3. As above, let us denote by Q, F the matrices representing the
Fourier transform of ∂/∂ q̄ and F (D). We have the following result:
Proposition 3.4. The pair (F, Q) forms a regular sequence in the ring R = Mat4 (R),
where R = C[x0 , . . . , x3 ].
Proof. The two matrices Q and F commute since the two operators of which they are
the Fourier transform commute. Let A be a matrix such that AQ−1 F ∈ R and recall
that Q−1 = (
P3
j=0
x2j )−1 Qt . The matrix F is the Fourier transform of the matrix
associated to the quaternionic polynomial F (q) that is regular (see for example [25])
so it cannot contain Q as a (left) factor. It follows that the only possibility is that A
contains Q as a factor, so that AQ−1 has polynomial entries.
We know that there is another natural class of differential operators acting on
70
regular functions: those are operators of the form
∂m
P (D) = m1 m2 m3 I4 ,
∂x1 ∂x2 ∂x3
m=
3
X
mi ,
i=1
where I4 is the 4 × 4 identity matrix. We have the following:
Corollary 3.5. Let us consider the system
½
P (D)f = g
Q(D)f = 0
where Q(D) is the Cauchy-Fueter operator and P is as above. Then the system has
a solution on open convex sets if and only if g is regular.
Proof. First, we show that P, Q satisfy the condition b) of Proposition 3.2. Let A be
a matrix such that M = AQ−1 P has polynomial entries. Note that if mij is the entry
of M at the place ij, we have
−1
mij = [AQ P ]ij =
X
k,l
=
X
k
aik
Qtkl
aik 2
plj =
(x0 + . . . + x23 )
X
Qtkj
Qtkj
a
p
=
p
,
jj
ik
2
2
(x20 + . . . + x23 )
(x
+
.
.
.
+
x
)
0
3
k
where p = p11 = · · · = p44 . Since p involves only xi with i = 1, 2, 3, it cannot have
factors in common with (x20 + x21 + x22 + x23 ). Then, in order for mij to be polynomial
it is necessary that [AQ−1 ]ij is a polynomial for every i, j.
We conclude with a problem involving the eigenvalue equation for the Dirac
71
operator, but the same can be shown for both the Cauchy-Fueter and the MoisilTheodorescu operators. Let us denote by T any of these three operators. Note that
T −λ factorizes the Helmholtz operators ∆−λ2 . We consider the problem: determine
a function f satisfying the system
½
T f = λf
p(T )f = g
(3.9)
where g is a given function in C ∞ , λ ∈ C is an eigenvalue, p is a polynomial with
complex coefficients and T denotes the conjugate of T .
Proposition 3.5. Let U be an open convex set in Rm . Then the problem (3.9) admits
a solution if and only if g is an eigenfunction of T related to the same eigenvalue.
Proof. Once again the proof is based on Proposition 3.2. We consider the Fourier
transform of the two equations in (3.9) and their representative matrices Q = T − λI,
(I the identity matrix) and P , which obviously commute. For any matrix A such
that AQ−1 P has polynomial entries, the matrix AQ−1 already has polynomial entries
in fact the elements in P cannot have any factor containing the term λ. Then the
syzygies of [Qt , P t ]t are [−P, Q] and the only compatibility condition on the datum g
is Qg = 0 which implies T g = λg.
Chapter 4: Biregular functions of several
quaternionic variables
4.1
Biregular functions
A full theory for functions of several quaternionic variables which satisfy the CauchyFueter operator is described in [25]. The results presented in the following sections are
a joint work with D.C. Struppa and I. Sabadini. Consequently some of the notation
has been adapted to follow the one used in our paper [30]. Recall from Chapter 2
that the left Cauchy–Fueter operator D` acting on infinitely differentiable functions
defined on the space H of quaternions is defined as
D` =
∂
∂
∂
∂
+i
+j
+k
,
∂x0
∂x1
∂x2
∂x3
where we are denoting by i , j , k the three imaginary units of the real associative
algebra of quaternions and a quaternion by q = x0 + i x1 + j x2 + k x3 . Since the
algebra of quaternions is non commutative, it is possible to move the imaginary units
to the right and define the right Cauchy-Fueter operator
Dr =
∂
∂
∂
∂
+
i+
j+
k.
∂x0 ∂x1
∂x2
∂x3
72
73
In section 2.3 I already presented some of the properties of the left Cauchy-Fueter
operator. It is well known that the theory of nullsolutions of the left or right CauchyFueter operators in one or several variables are completely equivalent. In this chapter,
we will instead study functions of an even number 2n of quaternionic variables which
are simultaneously left regular in the first n variables p1 , . . . , pn and right regular in
the remaining variables q1 , . . . qn . These functions, which are called biregular, are a
non-trivial generalization of functions of one or several quaternionic variables.
During the eighties, Brackx and Pincket have intensively studied biregular functions in two variables for Clifford valued functions, see [11–14]. In other words, they
were interested in the study of functions f : Rk × Rq → Rm , 1 < k, q < m, which are
left monogenic in one variable and right monogenic in the other one. Lots of results
were proved for this class of functions, for example the Cauchy Integral formulas,
the existence of the Taylor expansion (in terms of suitable homogeneous polynomials) and of the Laurent series, the Hartogs’ theorem on the removability of compact
singularities.
Our purpose is to generalize the study to functions which are biregular with respect
to several pairs of variables. The technique we used is more algebraic and will allow
to solve, in particular, the following problems:
a) find an explicit expression for the compatibility conditions of the system

D`1 (f )




 Dr1 (f )





=
=
···
D`n (f ) =
Drn (f ) =
g`1
gr1
(4.1)
g`n
grn
74
b) construct the free resolution for the associated module, finding the dimensions
of the free modules, the degrees of the maps and the length of the corresponding
complex
c) calculate the cohomology of this complex.
As a byproduct we will show that the Hartogs phenomenon holds for biregular functions in several variables (both in the regular and in the monogenic case) or rather
we will show that more general compact singularities can be eliminated. Note that
Hartogs’ phenomenon had already been proved in [12] for the case of n = 1.
4.2
Examples of computation in low dimension
Let us fix the necessary definitions and notations.
Definition 4.1. Let f : Hn × Hn → H be an infinitely differentiable function. The
function f (p1 , . . . , pn , q1 , . . . , qn ) is said to be biregular with respect to the pairs of
variables (ps , qs ), s = 1, . . . , n if and only if it satisfies the system

D`1 (f )




 Dr1 (f )





= 0
= 0
···
D`n (f ) = 0
Drn (f ) = 0
where ps = xs0 + ixs1 + jxs2 + kxs3 , qs = ys0 + iys1 + jys2 + kys3 and
D`s =
∂
∂
∂
∂
+i
+j
+k
,
∂xs0
∂xs1
∂xs2
∂xs3
Drs =
∂
∂
∂
∂
+
i+
j+
k.
∂ys0 ∂ys1
∂ys2
∂ys3
(4.2)
75
Any quaternionic equation D`s f = 0 or Drs f = 0 translates into four real equations
and can be written in matrix form as


∂xs0 −∂xs1 −∂xs2 −∂xs3
f0
 ∂xs1


∂xs0 −∂xs3
∂xs2   f1

 ∂xs2
∂xs3
∂xs0 −∂xs1   f2
∂xs3 −∂xs2
∂xs1
∂xs0
f3



∂ys0 −∂ys1 −∂ys2 −∂ys3
f0

 ∂ys1


∂ys0
∂ys3 −∂ys2   f1
 = 0, 

 ∂ys2 −∂ys3
∂ys0
∂ys1   f2
∂ys3
∂ys2 −∂ys1
∂ys0
f3


=0

respectively, where f has been considered as a vector with four real components.
In the sequel, we will denote by the symbols D`s and Drs both the operators and
the matrices associated to them. The context will make it clear the meaning of the
symbol. System (4.2) can be written in matrix form as the unique equation
Pn (D)f = 0.
We will be interested in the symbols of the system, i.e. in the matrix Pn obtained from
Pn (D) via Fourier transform. As usual in these cases, when we write the entries of Pn
we will neglect the imaginary unit
√
−1 and we will use the same variables instead of
dual variables. The main object of interest will be the cokernel of the map induced
by the matrix Pnt , i.e. the module Mn = R4 /hPnt i, R being the ring of polynomials in
8n variables R = C[xs0 , . . . , xs3 , ys0 , . . . , ys3 ], s = 1, . . . , n, and hPnt i being the module
generated by the columns of Pnt in the minimal free resolution of Mn (2.6).
In the next paragraphs I will use the computer package CoCoA to carry out some
computations for the cases of 1, 2, and 3 variables.
Case n = 1. Let us define the two 4×4 matrices representing the symbols of the right
and left operators, where we write p = x0 +ix1 +jx2 +kx3 and q = y0 +iy1 +jy2 +ky3 ,
thus avoiding the use of double indices, and we work with matrices with entries in
76
R = C[x0 , . . . , x3 , y0 , . . . , y3 ].
Use R::=Q[x[0..3]y[0..3]];
Dl:=Coala.FueterMat([x[0],x[1],x[2],x[3]]); -- left operator
Dr:=Coala.RightFueterMat([y[0],y[1],y[2],y[3]]); -- right operator
B:=Coala.BlockMat(Dl,Dr);
ModB:=Module(B);
--resolution for biregular functions
Res(R^4/ModB);
0 --> R^4(-2) --> R^8(-1) --> R^4
With the command Coala.Comm we can compute the commutator and check that it
is zero.
Coala.Comm(Dr,Dl);
Mat([
[0, 0, 0, 0],
[0, 0, 0, 0],
[0, 0, 0, 0],
[0, 0, 0, 0]
])
------------------------------Furthermore, denoting by R the ring of coordinates, the two matrices D` and Dr not
only commute but also form a regular sequence in Mat4 (R), since they involve two
different sets of variables. We can then apply Theorem 3.3 and conclude that the
77
syzygies associated to the module generated by the rows of the matrix P1 (D) =
¡D` ¢
Dr
are given by the matrix S1 (D) = (−Dr , D` ).
This is actually the last map of the free resolution of the module associates to
biregular functions of two quaternionic variable. The complex has a Koszul-like
1
form both in terms of Betti numbers and degrees.
P
S
1
1
0 −→ R4 (−2) −→
R8 (−1) −→
R4 −→ 0.
It is possible to prove that the second cohomology module is the only nonzero one.
Precisely, if we define by M1 the module associated to P1 ,
Ext2R (M1 , R) = R4 /Im(S1 )
while the first cohomology vanishes, as it can be easily checked with CoCoA , for
example by checking that the maximal minors of the matrix P1 are coprime (see [2],
lemma 1). This situation is again totally similar to the case of holomorphic functions
of two complex variables (though the real dimension in that case would be 2 instead
of 4). Because the Cauchy-Riemann operators in two complex variables commute
and form a regular sequence, the associated complex is again Koszul-like. We can
summarize these results in a proposition (the last part of which was essentially already
proved in [13]):
1
Saying that a complex has a Koszul-like form means that it looks like the one in condition 2) of
Theorem 3.4 and that its maps are constructed naturally from the blocks of the first matrix in the
complex.
78
Proposition 4.1. Let D` and Dr be respectively the left and the right Cauchy-Fueter
operators acting on functions f : H × H −→ H. Consider the inhomogeneous system
½
D` (f ) = g`
Dr (f ) = gr .
Then the only compatibility condition on the system is given by Dr g` = D` gr and the
associated complex is Koszul-like, i.e. it has length two and its maps are constructed
as in the Koszul complex. The Hartogs phenomenon holds for the solutions of the
system, while the second cohomology module of the associated complex is nonzero.
Case n = 2. The situation is completely different and much more intricate when we
consider functions of four quaternionic variables (p1 , q1 , p2 , q2 ) and we study functions
that are left regular in the p1 , p2 and right regular in q1 , q2 . Since every left operator
commutes with every right operator, we expect to obtain the Koszul syzygies of the
pairs (Dli , Drj ), for every i, j ∈ {1, 2}. On the other hand, the two left operators do
not commute with each other, and neither do the right operators. From the two pairs
(D`1 , D`2 ) and (Dr1 , Dr2 ) we then expect some syzygies of the same type appearing
in the complex associated to two left (resp. right) Cauchy-Fueter operators, which
are quadratic and come from the radial algebra (see Example 2.6).
Let P2 be the 16 × 4 matrix in R, representing these 4 operators. We can compute
the resolution of the associated module M2 = Cokernel(Pt2 ) using CoCoA . We obtain
0 --> R^4(-8) --> R^16(-7) --> R^16(-5)(+)R^16(-6) -->
--> R^40(-4) --> R^16(-2)(+)R^16(-3) --> R^16(-1) --> R^4
A we can see from the Betti numbers and the degrees of the maps in the resolution, the
first syzygies consists of 16 linear relations, corresponding to 4 quaternionic syzygies,
79
and 16 quadratic relations, corresponding to 4 quaternionic syzygies. The explicit
expression of such relations can be computed again with CoCoA, at least in their real
counterparts, and shows that the linear relations are Koszul and the quadratic ones
are exactly those described in [1] for the left operators and the corresponding ones
for the right operators. Furthermore CoCoA can help us calculate the cohomology
modules: they all vanish, except at the last spot of the complex where we have a
nontrivial cohomology, (the cokernel of the last map). The following proposition
summarizes the results for the case n = 2.
Proposition 4.2. Consider the inhomogeneous system

D`1 f



D`2 f
Dr1 f



Dr2 f
=
=
=
=
g`1
g`2
gr1
gr2
The compatibility conditions of the system are given by the following four (linear)
relations:
D`i grj = Drj gli ,
i, j ∈ {1, 2}
plus the four quadratic relations
Dli Dlj glj = Dl2j gli ,
i, j ∈ {1, 2}, i 6= j
Dri Drj grj = Dr2j gri ,
i, j ∈ {1, 2}, i 6= j.
The complex associated to the module M2 associated to the system is
P
S
S
S
S
S
2
1
2
3
4
5
0 −→ R4 −→
R16 −→
R32 −→
R40 −→
R32 −→
R16 −→
R4 −→ 0
80
where the self-duality condition holds on the maps of the resolution, i.e. S5 = P2t , S4 =
S1t and S3 = S2t . The complex is exact except at the last spot where the cohomology
module is the cokernel of S5 .
Proof. All the statements of the theorem can be easily checked with CoCoA. It is
immediate to show that the eight relations given for the system are compatibility
conditions. Their sufficiency follows from a dimension argument. The vanishing of
the Ext-modules Extj (M2 , R), j = 0 . . . 5 can be checked directly using CoCoA. The
last map is the only one that gives rise to a nontrivial cohomology.
Remark 4.1. In the case of left regular functions of four variables, the compatibility
conditions are quadratic and, besides the quaternionic ones, they include the so called
exceptional relations that involve operators that are different from the Cauchy-Fueter
(left or right) operator (see [4]). In this case, on the other hand, the complex for
biregular functions of four quaternionic variables has a double nature: it behaves
both like the Koszul complex and the Cauchy-Fueter one. Since the number of left
(or right) operators involved is only n = 2, we do not see any exceptional syzygies.
The exceptional behavior can occur only with at least three Cauchy-Fueter operators
of the same type (left or right).
Case n = 3. Before we generalize the results obtained so far for the case of n pairs
of quaternionic variables, let us show the computations of the complex for the case of
biregular functions of 6 quaternionic variables, defined as the kernel of 3 left operators
and 3 right operators. The following is the minimal free resolution of the module M3
(to get the number of quaternionic relations, it suffices to divide the Betti numbers
81
by 4) as obtained by CoCoA:
0 −→ R16 (−12) −→ R144 (−11) −→ R564 (−10) −→ H 1240 (−9) −→ R1620 (−8) ⊕ R48 (−7)
−→ R1200 (−7) ⊕ R232 (−6) −→ R400 (−6) ⊕ R432 (−5) −→ R360 (−4) −→
−→ R80 (−3) ⊕ R36 (−2) −→ R24 (−1) −→ R4 −→ M3 −→ 0
(4.3)
We notice immediately that the nice property of self-duality does not hold in this
case, since the Betti numbers are not symmetric. The length of the complex is
10, i.e. twice the length of the Cauchy-Fueter complex for 3 quaternionic variables.
This was the case even in the previous examples, and indeed we will show in the
next section this happens in general for functions on 2n variables. Let us now look
at the compatibility conditions. In the case of three left operators and three right
operators we have 9 linear quaternionic syzygies and 10 quadratic ones. This makes
us think that we can repeat the argument provided in the case of 4 variables in
order to count the first syzygies. Indeed, the 9 pairs of operators (Dli , Drj ) commute
and hence give rise to the 9 Koszul relations. The triple of left operators generates
¡¢
¡¢
10 = 2 32 + 4 33 quadratic relations, of which 2 are exceptional and 8 are purely
quaternionic, as described for example in [63]. The same holds for the set of right
operators, for a total of 20 quadratic relations. The relations described so far are
obviously syzygies, and it could be shown that they are independent exactly as it
has been done for the case of 3 Cauchy-Fueter operators and for the Koszul complex.
Therefore, for dimension reasons, they are all the syzygies and the only ones. As far
as the cohomology modules are concerned, the dimension of the problem in this case
has not allowed us to use CoCoA for the computation of Ext, or, equivalently, of the
82
minors of the symbol matrix. This is way too heavy for the machines that we are
using at present. Our guess would be that the complex is exact except at the last
point. The nature of the maps involved, in fact, seems to be the same as in the lower
dimensional cases.
4.3
Algebraic analysis of the module associated to
biregular functions
We present some preliminary lemmas that will lead to the proof of our results for the
general case of the module associated to n left operators and n right operators.
Lemma 4.1. Let A1 , . . . , An and B be square matrices representing n + 1 linear constant coefficient differential operators. Let us suppose that Ai B = BAi for every
i = 1, . . . , n and suppose that they form a left regular sequence in the ring of matrices. Let S = {(Sj1 , . . . , Sjn )|j = 1 . . . t} be a set of generators for the module
of left syzygies of the n-tuple (A1 , . . . , An ). Then the module Syz(A1 , . . . , An , B) is
generated by the set S 0 = {(Sj1 , . . . , Sjn , 0)|j = 1 . . . t} together with the set K =
{(0, . . . , −B, . . . , 0, Ai )|i = 1 . . . n}.
Proof. It is immediate to see that the elements of S 0 and K are syzygies. Let us now
show they are sufficient to generate them all. Let C1 , . . . , Cn , D be n + 1 matrices
such that
C1 A1 + · · · + Cn An + DB = 0.
Then from C1 A1 + · · · + Cn An = −DB and the fact that (A1 , . . . , An , B) is a left
regular sequence it follows that D = T1 A1 + · · · Tn An for some matrices T1 , . . . , Tn .
83
By substituting this expression of D and by using the commutativity we have that
(C1 + T1 B)A1 + · · · + (Cn + Tn B)An = 0
and so Ci + Ti B = P1 S1i + · · · Pt Sti for every i = 1 . . . n and so we get that the
n + 1-tuple (C1 , . . . , Cn , D) is of the desired form.
Remark 4.2. One may think that Lemma 4.1 can be generalized to the case of any
number of operators, saying that the syzygies of two sets of operators A1 , . . . , An
and B1 , . . . , Bm such that Ai Bj = Bj Ai for each i, j can be reconstructed from
Syz(A1 , . . . , An ) and Syz(B1 , . . . , Bn ) and from the commutation relations only. Although the following results will prove that this is true for the case of the system
describing biregular functions in 2n quaternionic variables, this is not true in general. As a counterexample, consider an associative algebra generated by 4 elements
A1 , A2 , B1 , B2 with the following defining relations:
A2i = 1,
Bi2 = −1,
Ai Bj = Bj Ai ,
i, j = 1, 2.
It is then clear that we can compute the syzygies of (A1 , A2 ) and (B1 , B2 ) separately
as follows:
Syz(A1 , A2 ) = h(A1 , −A2 )i,
Syz(B1 , B2 ) = h(B1 , −B2 )i,
but the elements (A1 , 0, B1 , 0) and (0, A2 , 0, B2 ) are clearly syzygies of the tuple
(A1 , A2 , B1 , B2 ) which are not generated by the natural syzygies {(A1 , −A2 , 0, 0),
(0, 0, B1 , −B2 ), (−B1 , 0, A1 , 0), (−B2 , 0, 0, A1 ), (0, −B2 , 0, A2 ), (0, −B1 , A2 , 0)}. The
above algebra can be generated, for example, as the subalgebra of Mat2 (C) generated
by A1 =
¡1
0
0 −1
¢
¡ ¢
¡ ¢
¡ ¢
, A2 = 10 10 , B1 = i0 −i0 , B2 = i0 0i .
84
The following lemma exploits the computation of the Gröbner Basis of the module
associated to biregular functions for small values of n to infer the general case. We
always assume that the default term ordering on the ring C[xi0 , . . . , xi3 , yi0 , . . . , yi3 ]
is DegRevLex.
Lemma 4.2. Let D`1 , . . . , D`n be the symbols matrices associated to n left CauchyFueter operators and let Dr1 , . . . , Drn be the symbols of n right Cauchy–Fueter operators. Let Bn be the module generated by the rows of such matrices. The reduced
Gröbner Basis for Bn is given by the rows of the matrices D`s and Drs , i = 1, . . . , n
together with the rows of the matrices
Bks = D`k D`s − D`s D`k
and
Cks = Drk Drs − Drs Drk ,
1 ≤ k < s ≤ n.
Proof. The statement can be verified directly with CoCoA for n ≤ 4. For the general
case, we can see that the S-polynomials generated by any two rows of D`i give rise
to the rows of Bks and the S-polynomials of two rows of Dri generate the rows of
Cks . If we pick a row of a D`i and a row of a Drj , their S-polynomial reduces to
zero due to the commutativity D`i Drj = Drj D`i . Therefore, Buchberger’s algorithm
generates new elements of the Gröbner Basis of Bn by adding the rows of Bks and Cks
to the original ones. To prove that they are the only elements of the reduced Gröbner
Basis, we need to show that all their S-polynomials reduce to zero. An S-polynomial
generated by a row of D`i and a row of Bks is computed and reduced to zero as in
the case n = 2 or n = 3, depending on the number of different indices in the triple
(i, k, s). The same holds for Dri and Cks . An S-polynomial generated by two rows of
Bks is computed and reduced as in the case n = 2, n = 3 or n = 4 depending on the
number of different indices. The same holds for two rows of Cks . When choosing a
85
row of D`i and a row of Cks , or a row of Dri and a row of Bks , or a row of Bks and a
row of Ctl , the commutativity of left operators with right operators implies that the
S-polynomials are identically zero.
The next step is to calculate the Hilbert series. The following lemma provides the
series for the general case of n > 2.
Lemma 4.3. Let R = C[xi0 , . . . , xi3 , yi0 , . . . , yi3 ] and let Mn = R4 /Bn be the R-module
associated to n left Cauchy-Fueter operators and n right Cauchy-Fueter operators,
with n > 2. Then the Hilbert series of the module Mn is given by
HMn (t) = 4
(1 + (n − 1)t)2
.
(1 − t)4n+2
Moreover, the module is Cohen-Macaulay.
Proof. Let us first calculate the monomial module LTBn . Computations with CoCoA in
the case n = 3 and n = 4 show that it is generated by the set
{xi0 et , yi0 et , xh2 xk1 et , yh2 yk1 et | i = 1 . . . n, t = 1 . . . 4, 1 ≤ h < k ≤ n}.
The same argument as in Lemma 4.2 show that this is sufficient to characterize
the module. Let In be the ideal Ix + Iy = {xi0 , xh2 xk1 | i = 1 . . . n, 1 ≤ h < k ≤
n} + {yi0 , yh2 yk1 | i = 1 . . . n, 1 ≤ h < k ≤ n}. Then obviously HMn (t) = 4HR/In (t).
Since Ix and Iy involve a different set of variables, we can use the isomorphism
R/In ' R/Ix ⊗ R/Iy
and conclude that HR/In = HR/Ix · HR/Iy . Since HR/Ix = HR/Iy =
1+(n−1)t
(1−t)2n+1
as
calculated in [3], we find the final form of the series as wanted. It follows that the
86
dimension of the module Mn is 4n + 2 and to prove that it is Cohen-Macaulay we
have to show that depth(Mn ) =dim(Mn ). A maximal regular sequence for Mn can be
constructed as a maximal regular sequence for R/In . The latter has obviously twice
the number of elements than the one constructed in [3] for R/Ix , which has length
2n + 1, so the statement follows.
Being able to calculate the monomial module LT(Bn ), as in the proof of the above
proposition, allows to give an explicit formula for the Betti numbers of the minimal
free resolution of Mn . Note that LT(Bn ) is ”diagonal” in the sense specified below.
The property of LT(Bn ) being diagonal translates into the fact the Betti numbers
of Mn are exactly the ones of the R/In multiplied by 4. Moreover, In splits into
Ix + Iy , which are the ideals in the diagonal of Bx and By (with obvious meaning of
the symbols). Then the resolution of Mn is the ”product” of the resolutions of Mx
and My , in the following sense:
Definition 4.2. Let R be a ring and I and J two ideals of R. Let {(Fi , ϕi )}i and
{(Gj , ψj )}j be the two minimal free resolutions of R/I and R/J respectively. In view
of the R-modules isomorphism
R/(I + J) ' R/I ⊗ R/J
(4.4)
we can define the tensor product resolution of R/(I + J) as the complex {(Td , τd )}d
where
Td =
M
i+j=d
Fi ⊗ Gj ,
τd =
X
ϕi ⊗ ψj .
i+j=d
Given the isomorphism (4.4) the tensor product resolution is indeed a free resolution for the quotient R/(I + J). Its minimality follows form the definition of the
87
maps τd and form the fact that the matrices in the free resolutions of R/I and R/J do
not have nonzero constant entries. Let us now state a proposition that generalizes,
under suitable hypotheses that can be easily checked, the construction of a tensor
product resolution to the case of modules. Given an ideal I of R, we denote by ∆s (I)
the diagonal submodule of Rs given by Ie1 ⊕ · · · ⊕ Ies where ei is the element of the
canonical basis of Rs .
Proposition 4.3. Let s be an integer, let R be a polynomial ring and let M1 , M2 and
M = M1 + M2 be finitely generated submodules of the free module Rs . Let I1 and I2
be two ideals in R such that:
1) LT(M ) =LT(M1 )+LT(M2 ),
2) LT(Mi ) = ∆s (Ii ), i = 1, 2
and let αi and βj be the Betti numbers associated to M1 and M2 respectively. Then
the Betti numbers of the module M are given by
γd =
1 X
αi βj .
s i+j=d
(4.5)
Proof. Let αi0 and βj0 be the Betti numbers of the resolutions associated to R/I1 and
R/I2 . From condition 1) and 2) it follows that LT(M ) = ∆s (I), where I = I1 + I2 ,
so if γd0 are the Betti numbers of R/I, we have that
γd = sγd0 ,
αi = sαi0 ,
βj = sβj0 ,
for all i, j, d ∈ N.
By definition of tensor product resolution, we obtain the Betti numbers associated
P
0
0 0
to R/I via the formula γd0 =
i+j=d αi βj so substituting in γd = sγd we get the
statement.
88
Remark 4.3. The expression for the Betti numbers given in (4.5) does not take into
account the degrees of the maps involved in the resolution. However, one could extend
the results from proposition 4.3 giving a formula for the graded Betti numbers. This
requires the tensor product resolution to be endowed with the natural grading arising
form the tensor products. Denoting by γd (λ) the d-th Betti number in degree λ of
the module M , formula (4.5) becomes, with obvious meaning of symbols
γd (λ) =
X
1
αi (%)βj (σ).
s i+j=d, %+σ=λ
We can now state all the principal results for the analysis of the module Mn in a
Theorem:
Theorem 4.1. Let n > 2, R = C[xi0 , . . . , xi3 , yi0 , . . . , yi3 | i = 1 . . . n] and consider the
system associated to n left Cauchy-Fueter operators D`1 , . . . , D`n and n right CauchyFueter operators Dr1 , . . . , Drn . Let Mn be the R-module associated to the map given
by all the 2n operators. Then the length of the minimal free resolution of Mn is 4n − 2
and the complex is exact except at the last point. The Betti numbers associated to Mn
are γ0 = 4, γ1 = 8n and
X µ2n − 1¶µ2n − 1¶ ij + 1 − d
,
γd = 4n
j
i
ij + 1 + d
i+j=d
2
d > 1.
(4.6)
Furthermore, if we consider the inhomogeneous system

D`1 (f )




 Dr1 (f )





=
=
···
D`n (f ) =
Drn (f ) =
g`1
gr1
(4.7)
g`n
grn
89
the compatibility conditions are given by the n2 linear relations
D`i grj = Drj g`i ,
i, j ∈ {1, . . . , n}
(4.8)
¡ ¢
¡ ¢
and the 4 n2 + 4 n3 relations given by the following
2
gsi ,
Dsi Dsj gsj = Dsj
i, j ∈ {1, . . . , n}, i 6= j
Dsi Dsj gsk + Dsj Dsi gsk = Dsk Dsi gsk + Dsk Dsj gsi ,
(4.9)
i, j ∈ {1, . . . , n}, i 6= j
¡ ¢
and finally the 4 n3 exceptional relations
0
0
0
0
0
0
(Dsi
Dsj −Dsj
Dsi )gsk +(Dsj
Dsk −Dsk
Dsj )gsi +(Dsk
Dsi −Dsi
Dsk )gsj = 0,
1≤i<j<k≤n
(4.10)
00
00
00
00
00
00
(Dsi
Dsj −Dsj
Dsi )gsk +(Dsj
Dsk −Dsk
Dsj )gsi +(Dsk
Dsi −Dsi
Dsk )gsj = 0,
1≤i<j<k≤n
where in each line Ds stands for either the left operator of the right operator, and the
operators Ds0 and Ds00 are Cauchy-Riemann like operators involving only two of the
four real variables corresponding to the quaternionic variable given by the index (see
[4] for their explicit form).
Proof. The length of the resolution follows form the Auslander-Buchsbaum formula
and from Lemma 4.3:
pd(Mn ) = 8n − depth(Mn ) = 8n − (4n + 2) = 4n − 2.
The vanishing of the cohomology modules follow from the computation of the dimension of the characteristic variety, which will be presented in a subsequent theorem.
Consider the sum Bn = Bx + By , where Mx is the module associated to the left operators and My to the right operators. From the calculation of the leading term module
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done in Lemma 4.3 we have that ∆(In ) =LT(Bn ) =LT(Bx )+LT(By ) = ∆(Ix ) + ∆(Iy )
so we can apply Proposition 4.3 to calculate the Betti numbers for Mn . Mx and My
have the same Betti numbers, as calculated in [3]: αi = βi = 4
¡2n−1¢ n(i−1)
i
i+1
so formula
(4.6) follows from the expression of the Betti numbers given in Proposition 4.3. Let
us now focus on the computation of syzygies. Let us denote by D`i the symbol of
the operator D`i and by Dri the symbol of Dri . Using Lemma 4.2 we know that
the reduced Gröbner Basis of the module associated to the system (4.7) consists of
vectors of degree at most two, involving two different sets of variables xi or yi at a
time. The algorithm to compute the syzygies of a module (see [54]) consists in the
computation of an S-polynomial for each pair of elements of a Gröbner Basis and
then in their reduction modulo Bn using the division algorithm. Since this last step
cannot involve variables not already in the S-polynomial, each syzygy will contain at
most 4 different sets of variables, hence it will involve at most four different operators.
Direct computations for the case n = 4 show that in reality the number of operators involved in each syzygy is at most three. Let us then consider three operators
(A1 , A2 , B) among those appearing in the system. If they are all left Cauchy-Fueter or
all right Cauchy-Fueter, the syzygies arising will be of type (4.9) or (4.10). If A1 and
A2 are left operators and B is a right operator (or viceversa), we can apply Lemma
4.1 because the three operators involve different variables and Ai commutes with B.
We then conclude that the syzygies are either Koszul (4.8) or of the ”Cauchy-Fueter”
types (4.9) or (4.10).
Remark 4.4. It is possible to calculate the graded Betti numbers using the formula
of Remark 4.3 and the graded Betti numbers of the Cauchy–Fueter complex. For the
sake of brevity, we refer the reader to our webpage [28] where an explicit expression
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is given.
Remark 4.5. Note that all the examples in Section 4.2 are an immediate consequence
of Theorem 4.1 and Corollary 4.1.
The exactness of the complex associated to biregular functions in 2n quaternionic
variables depends on the vanishing of the Ext-modules Extj (M, R). To show that
those modules vanish for j = 0, 1, . . . , 4n − 3 we use the fact that the characteristic
variety associated to Mn (essentially the affine variety of points in which the rank of
the matrix Pn is strictly less than 4) has codimension 4n − 2 and a well known result
in [60] (Proposition 2, p. 139).
Theorem 4.2. The characteristic variety Vn associated to Mn has dimension 4n + 2.
Proof. Let us consider the complexified algebra of quaternions HC = H ⊗ C. A
quaternion ξi = ξi0 + ξi1 i + ξi2 j + ξi3 k will also be denoted as a column vector in C4 :
ξi = (ξi0 , ξi1 , ξi2 , ξi3 ). The conjugate of a quaternion ξi will be the element ξi∗ = ξi0 −
ξi1 i −ξi2 j −ξi3 k and will be associated to the column vector ξi∗ = (ξi0 , −ξi1 , −ξi2 , −ξi3 ).
We will show that the algebraic set Vn has dimension 4n + 2 in a neighborhood of
an arbitrary point in Vn . An element in Vn is ζ = (p1 , . . . , pn , q1 , . . . , qn ) where
pi , qi ∈ C4n , i = 1, . . . , n. Note that the columns of the matrix Pnt correspond to the
quaternions (cfr. [4] and [25])
p∗1 , p∗1 i , p∗1 j , p∗1 k , . . . , p∗n , p∗n i , p∗n j , p∗n k , q1∗ , i q1∗ , j q1∗ , k q1∗ , . . . , qn∗ , i qn∗ , j qn∗ , k qn∗ .
The determinant of the i-th 4 × 4 block in Pnt is equal to (p∗i pi )2 if i ≤ n or to (qj∗ qj )2
if i = n + j. The equation ξi∗ ξi = 0 defines a quadratic cone V of dimension three in
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C4 . Now for ξ ∈ HC , we define four complex subspaces of HC as follows:
Lξ = {ξq | q ∈ HC },
L⊥
ξ = {q ∈ HC | : ξq = 0}
Rξ = {qξ | q ∈ HC },
Rξ⊥ = {q ∈ HC | qξ = 0}.
and
The spaces Lξ and Rξ are the image of left and right multiplication by ξ respectively,
while the other two spaces are the kernels of these maps. By consequence
⊥
dimC Lξ + dimC L⊥
ξ = dimC Rξ + dimC Rξ = 4.
It is known (see [4] and [25]) that if ξ ∈ V1 and ξ 6= 0 then ξ ∗ ξ = 0 and dimC Lξ +
dimC L⊥
ξ = 2, (in fact the map of the left multiplication by ξ corresponds to the first
four columns of Pnt with ξ ∗ substituted in it). It is easy to verify, for example using
CoCoA, that the 3 × 3 minors of this matrix are multiples of ξ ∗ ξ and the fact that
ξ 6= 0 implies that not all the 2 × 2 minors are zero. Since Lξ ⊆ L⊥
ξ ∗ , as a consequence
⊥
of the dimension we get Lξ = L⊥
ξ ∗ and similarly dimC Rξ = 2 and Rξ = Rξ ∗ .
We now prove the following:
ζ ∈ V (Mn ) ⇐⇒ p1 , q1 ∈ V and pj ∈ Rp1 , qj ∈ Lq1 , j = 2, . . . n.
Let us prove the implication ⇐. If p1 ∈ V and pj ∈ Rp1 for j = 2, . . . n then pj = p0j p1
for a suitable p0j ∈ HC , therefore p∗j e ∈ Lp∗1 where e = 1, i, j, k so that the space
generated by the first n columns is contained in the two dimensional space Lp∗1 . In an
analogue way, if q1 ∈ V and qj ∈ Lq1 then the space generated by the last n columns
is contained in the two dimensional space Rq1∗ . The rank of Pnt is not maximum and
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so ζ ∈ Vn . Let us prove the converse. Let us suppose that ζ ∈ Vn . Then the matrix
Pnt is not of maximal rank and consequently the determinant of the first 4 × 4 block is
zero. This corresponds to p1 ∈ V and since dimC Lp∗1 = 2 we may assume that p∗1 and
p∗1 i form a basis for Lp∗1 . Moreover, the hypothesis on the rank of Pnt implies that, for
any fixed `, the elements p∗1 , p∗1 i, p∗` , p∗` i are linearly dependent. Reasoning as in [4],
we deduce that
p∗` ∈ L⊥
p1 = Lp∗1 .
We conclude that p∗` = p∗1 p0 for some p0 ∈ HC thus p` ∈ Rp1 . We now look at the last
n blocks of the matrix. It is obvious that q1∗ ∈ V . Note that by adding the columns
corresponding to the quaternions qj to the columns corresponding to the quaternions
pi , the space of columns does not have dimension two anymore. For example, if we
consider the elements p∗1 , p∗1 i, q1∗ , iq1∗ we have that they are linearly dependent, but the
rank of the 4 × 4 matrix they form is three. Nevertheless, we have that if q1 ∈ V then
dimC Rq1∗ = 2 and we may assume that q1∗ and iq1∗ form a basis for Rq1∗ . Moreover,
the fact that the rank of Pnt is not maximal implies that, for any fixed `, the elements
q1∗ , iq1∗ , q`∗ , iq`∗ are linearly dependent so for any choice of ` we still have that the
columns form a two dimensional subspace. In fact, as before we deduce that
q`∗ ∈ Rq⊥1 = Rq1∗ .
It follows immediately that
dim (V (Mn )) = 2dimC V +(n−1) dimC Rp1 +(n−1) dimC Lq1 = 6+4(n−1) = 4n+2.
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As a consequence of the theorem we obtain
Corollary 4.1. If Mn is as above, then we have
Exti (Mn , R) = 0,
for all
i = 0, . . . , 4n − 3
and
Ext4n−2 (Mn , R) 6= 0.
Proof. Since the characteristic variety V (Mn ) has dimension 4n + 2, we immediately
obtain that Exti (Mn , R) = 0, for all i = 0, . . . , 8n − (4n + 2) − 1.
Chapter 5: Invariant operators and the
Cauchy-Fueter complex
In this chapter I would like to suggest an alternative approach to the one followed for
the algebraic analysis of the operators described in the previous chapters. Following
some recent results (see [5, 51, 67]) it is possible, at least in principle, to construct
a Dolbeault sequence for the module associated to a differential operator without
the use of syzygies. On the other hand, Algebraic Analysis methods offer a general
construction of a resolution of a given overdetermined differential operator. In the
case of higher dimension, there are formidable computational problems connected
with this general construction. An alternative is based on symmetry considerations
and representation theory. In [26] the authors show that if the operator to be resolved
has a known symmetry, it is possible to use this information to efficiently reduce the
computational complexity of the problem. This way, one obtains a sequence that
is actually isomorphic to the one constructed with syzygies, at least in the case of
the Cauchy-Fueter operator. A more explicit construction of the syzygies using this
approach is presented in [18]. This chapter collects the results from both papers and
expands some of the computational aspects.
The main idea to construct an invariant resolution can be expressed as follows.
In general, if the first operator in the sequence is invariant with respect to a given
action, the same can be assumed also for all the operators appearing in complex. To
apply this idea to the construction of higher dimensional analogues of the Dolbeault
95
96
sequences, one has to find a symmetry for the operator defined by several Dirac
derivatives. We are going to study this question in real dimension four, i.e. in the
quaternionic case.
Quaternionic geometry is a special case of the so called parabolic geometries. A
short review of parabolic geometries is given in the next section. An extensive series of
complexes composed by invariant differential operators was constructed by R. Baston
in [5]. Such a construction is tightly related to the resolution constructed in [2–4]
and the form of the resolution can be efficiently deduced for any number of variables.
We will give the proofs of these facts without the use of the heavy machinery and
technicalities appearing in [5], and by cutting the number of computations implied in
[2–4] by a factor of two.
5.1
Parabolic Geometries
The notion of parabolic geometry was introduced in [43], following the Fefferman
concept of parabolic invariant theory in [40,41]. These geometries in their flat version
go back to Klein’s definition of geometry as the study of homogeneous spaces G/P.
Parabolic geometries are modeled on a homogeneous space M = G/P, where G is a
semisimple Lie group and P its parabolic subgroup.
Cartan then created a curved version of the geometry in question on a manifold
M. In this curved version, he replaced the principal fiber bundle p : G → G/P of the
homogeneous model with a general principal fiber bundle G → M with fiber group
P and he replaced the Maurer-Cartan connection (which is a one-form on G with
values in g, its Lie algebra) by a one-form ω on G with values in g, having suitable
properties deduced from the properties of the Maurer-Cartan form. This form ω on
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G is called the Cartan connection. It plays, in a sense, the role of the Levi-Civita
connections in Riemaniann geometry. The couple (G, ω) is then called a parabolic
structure on the manifold M, modeled on the homogeneous space G/P. The chosen
manifold M can have several different parabolic structures. For example, the sphere
can be considered as a manifold with a given projective, conformal, or quaternionic
structure. A nice introduction to parabolic geometries can be found in [64].
For our purposes, it is sufficient to consider homogeneous models M = G/P with
the corresponding Maurer-Cartan form. Furthermore, we will work only on a big cell
inside M. A big cell in M is a vector space V embedded in M, which is an open and
dense subspace of M. In conformal geometry (which is the most typical example of
a parabolic geometry), M is a sphere S n of dimension n and V is Rn embedded into
S n by stereographic projection.
We will consider here another example of a parabolic geometry, the so called
quaternionic geometry. Its homogeneous model is the quaternionic projective n-space
Pn (H) and the big cell inside is the space Hn of n quaternionic variables embedded
into Pn (H).
For each linear representation E of the (parabolic) structure group P, there is the
associated homogeneous vector bundle E(G/P ) over the corresponding homogeneous
space G/P. The bundle E(G/P ) is defined as the quotient G × E/ ∼, where the
equivalence relation is defined by
(g, e) ∼ (gp, p−1 e); g ∈ G, e ∈ E, p ∈ P.
There is a natural action of the group G on the vector bundle E(G/P ), induced by
the left action g 0 · (g, e) = (g 0 g, e); g, g 0 ∈ G, e ∈ E. Hence there is also the induced
98
action of G on sections of bundles E(G/P ). Invariant operators on M = G/P are then
defined as those operators on such sections, which commute with the above actions.
We will consider only invariant differential operators. When working on the big cell,
vector bundles in question are trivial bundles V × E.
One important fact to realize is that such invariant differential operators are rare
beings and that their classification is known in many cases, if the operators act between bundles associated to irreducible representations of P. The classification was
found using tools of representation theory (Verma modules and their morphisms, see
[9, 10] ). Irreducible representations of P coincide with irreducible representations of
the Levi factor G0 of P. The Levi factor G0 is a reductive group and its irreducible
representations are well understood.
Consider differential operators from C ∞ (V, E1 ) to C ∞ (V, E2 ). An invariant differential operator is characterized by a choice of its source and target (i.e., by a choice
of irreducible representations E1 and E2 ) up to a constant multiple. But for most of
the choices of E1 and E2 , there is no invariant differential operator at all! Irreducible
representations of G0 are classified by their highest weights. A very useful fact is that
such invariant differential operators can act only between points of an orbit of a finite
group on the space of weights. Hence the choice of values for invariant operators is
also enormously constrained. We will describe these facts in more details in the case
of quaternionic geometry. Similar facts for other parabolic geometries can be found
in [6].
We will encounter here only invariant differential operators of order one and two.
A general description of first order invariant differential operators on any manifold
with a given parabolic geometry can be found in [65]. Similar questions are much
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more complicated for higher order operators. A description of a large class of such
invariant operators can be found in [19]. We will need here only the relatively simple
cases of fist order and second order operators.
5.1.1
Invariant first order operators
We want to describe explicitly the form of first order invariant systems of differential
operators for flat models of parabolic geometries (see [65]). Suppose that G is a
real semisimple Lie group and P its parabolic subgroup. Let M = G/P be the
corresponding homogeneous space. We will treat only the |1|-graded case (for more
general cases, see [65]).
We will suppose that the Lie algebra g of G has a grading g = g−1 ⊕ g0 ⊕ g1 , where
g0 is the Lie algebra of the Levi factor G0 of P. The sum p = g0 ⊕ g1 is a parabolic
subalgebra of g, namely the Lie algebra of P. The subalgebra g−1 can be considered
as a representation of P by identification g−1 ≡ g/p. The vector space V = g−1 can
be embedded (using the exponential map) to M as a big cell. Moreover, the tangent
space T M is the associated vector bundle corresponding to the module g−1 . Similarly,
the P -module g1 is a model for the cotangent bundle, meaning that the cotangent
bundle T ∗ M is associated to the P -module g1 .
The group G0 is reductive and can be written as a product of a semisimple part
Gs0 and a commutative group R+ . Hence every irreducible representation of G0 is
the tensor product of a (complex) one dimensional representations of R+ and an
irreducible representation of Gs0 . The first is specified by a real number w and consist
in the multiplication by λw . We will denote it by Cw (this is a generalization of a
conformal weight from the case of conformal geometry). The second is specified by
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its highest weight.
Suppose that E is an irreducible Gs0 -module and Cw , w ∈ R is the one dimensional
representation of R+ on C, . Then E(w) will denote an irreducible G0 (hence P )module E⊗Cw . Any irreducible P -module can be written in such a way. A comfortable
way to encode both pieces of information for such P -modules is to use the weight λ
in the dual of the Cartan subalgebra of the whole Lie algebra g. We will describe this
below in more details for quaternionic geometry.
Let us now consider first order invariant differential operators between smooth
maps defined on domains in V = g−1 with values in the Gs0 -module E. The weights w
will be specified later. Suppose that E is given. Then there is only a finite number of
possibilities for invariant first order differential operators acting on such values. They
are all constructed by the following procedure.
Take f ∈ C ∞ (V, E). Then its gradient ∇f belongs to C ∞ (V, g1 ⊗ E). It is possible
to show that the product g1 ⊗ E of Gs0 -modules decomposes in a unique way into
irreducible components
g1 ⊗ E = F1 ⊕ . . . ⊕ Fk ,
where there are no multiplicities in the decomposition. Denote by πi the natural
projections of g1 ⊗ E to Fi , i = 1, . . . , k.
Then for every i = 1, . . . , k there exists a unique number wi ∈ R such that the
composition Di := πi ◦ ∇ is an invariant first order differential operator mapping
smooth maps with values in E(wi ) into smooth maps with values in Fi (wi − 1). Any
other invariant first order operator on sections of E is isomorphic to an operator of
the type Di . So we see that to find all first order operators, it is necessary only to
be able to decompose the tensor product of an irreducible Gs0 -module into irreducible
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components (one-dimensional representation of the commutative part play no role in
the decomposition). There are standard techniques available for such decompositions
and the result is known in all cases.
5.1.2
Certain invariant second order operators
A description of invariant second order operators is a much more complicated question. There are certain constructions available for higher order invariant operators
(see e.g. [19]) but they do not cover the case we need.
On the other hand, we can often find suitable candidates by following a procedure
similar to the one used above. We will describe it now. Suppose that we would like
to construct an invariant second order operator on functions on V with values in an
irreducible Gs0 -module E. Let us consider again the splitting
g1 ⊗ E = F1 ⊕ . . . ⊕ Fk
and similarly, decompose also products
g1 ⊗ Fi = Fi1 ⊕ . . . ⊕ Fili .
Then
g1 ⊗ g1 ⊗ E = ⊕ij Fij
(5.1)
is a decomposition of the left hand side into irreducible pieces. This time, however,
there can be higher multiplicities, i.e. certain summands can be isomorphic as G0 modules, while the other ones appear in (5.1) with multiplicity one. If Fij is such
a summand, and πij is the corresponding invariant projection, the operator f →
πij (∇∇f ) is the only possible candidate for an invariant second order operator from
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sections of E to sections of Fij . The remaining question is whether a number w in the
definition of E can be chosen in such a way that the operator is invariant, but we will
not discuss this question in details as it would lead us beyond our original scope.
5.2
Orbits of the Weyl group in the weight space
As explained above, irreducible representations of G0 can be characterized by a weight
λ in the dual h∗ of the Cartan subalgebra of g. A general statement (proved by
representation theory) says that an invariant operator from C ∞ (V, Eλ ) to C ∞ (V, Eµ )
can exist only if both weights are on the same orbit of the affine Weyl action on h∗ .
Due to the fact that W is a finite group, it gives just a finite number of candidates
for a given Eλ . We will describe it in more details below in the case of interest.
5.2.1
Quaternionic geometry
Our setting for invariant quaternionic complexes is the quaternionic geometry. The
group G is the projective group of quaternionic projective geometry Pn (H). This is
the quotient of the space Hn+1 \ {0} by the equivalence relation
(q0 , . . . , qn ) ≡ (q0 r, . . . , qn r), r ∈ H \ {0}.
The group of GLn+1 (H) of invertible quaternionic matrices is acting on Pn (H) in an
obvious way. The action has a kernel and the quotient of GLn+1 (H) by this kernel
is the group of projective transformations G. Its Lie algebra is the vector space of
(n + 1) × (n + 1) quaternionic matrices with the zero trace.
The grading g = g−1 ⊕ g0 ⊕ g1 is given by a block decomposition of g induced by
the decomposition of Hn+1 = H ⊕ Hn . The diagonal part is the Lie algebra g0 , while
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strictly lower triangular matrices form a commutative algebra g−1 and strictly upper
triangular matrices form a commutative algebra g1 . Multiplication of matrices shows
immediately that the decomposition is indeed a grading.
The Cartan subalgebra h ⊂ g can be chosen to be the diagonal subalgebra
{H ∈ g| H = diag(q0 . . . , qn ),
n
X
qi = 0, qi ∈ C ⊂ H},
0
where C ⊂ H is given by quaternions of the form q = x0 + ix1 . It is clearly a maximal
commutative subalgebra of g and h ⊂ g0 .
All representations we will consider will be complex representations. Hence they
will be at the same time representation of the complexification gC0 . This Lie algebra g0
is a sum g0 = sl1 (H) ⊕ sln (H) ⊕ R and its complexification gC0 is a reductive complex
Lie algebra equal to the sum sl2 (C) ⊕ sl2n (C) ⊕ C.
The complexification hC is a Cartan subalgebra in gC and hC ⊂ gC0 . Any irreducible
representation of gC0 can be written as a tensor product of an irreducible representation
of sl2 (C) with an irreducible representation of sl2n (C) and with a one-dimensional
representation of the commutative part C.
Suppose that ωi ∈ (hC )∗ , i = 1, . . . , 2n − 1 are fundamental weights for the Lie
algebra sl2n (C). Then any irreducible representation of the reductive Lie algebra gC0
can be uniquely characterized by its highest weight λ =
P2n−1
i=1
λi ωi with λi ∈ Z, λi ≥
0; i = 1, 3, 4, . . . , 2n − 1 and λ2 ∈ C. Hence we will use the sequence of coefficients
(λ1 , λ2 , λ3 , . . . , λ2n−1 ) as the label of the corresponding irreducible g0 -module.
To give the most important examples, the defining representation E ' C2 of sl2 (C)
corresponds to the highest weight (1, 0, 0, . . . , 0). The weight (j, 0, . . . , 0) correspond
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to the symmetric power ¯j (C2 ) of the previous representation.
The weight (0, 1, 0, . . . , 0) corresponds to a one-dimensional representation of the
commutative factor of g0 . We will denote it by C[1], its dual by C[−1] and their
powers by C[k], k ∈ Z. A tensor product of a representation V with C[−k] will be
denoted by V[−k].
The weights (0, . . . , 0, 1, 0, . . . , 0) with 1 at the (k + 2)-th place correspond to the
outer power Λk (C2n ) of the defining representation of sl2n (C).
Definition 5.1. Let W be the Weyl group of the Lie algebra sl2n+2 (C). This is a
finite group generated by reflections in (hC )∗ . Let δ =
P2n−1
i=1
ωi be the sum of all
fundamental weights. The affine action of an element w on (hC )∗ is defined as
w · λ := w(λ + δ) − δ.
The general theory on invariant operators tells us that if there is an invariant
operator (for a given parabolic geometry) from maps with values in Eλ to maps with
values in Eµ , than both highest weights λ, µ ∈ (hC )∗ should belong to the same orbit
of the affine action of the Weyl group. Hence we have always a finite number of such
weights.
An orbit of the Weyl group W is called regular, if the action of W on the orbit is
free. The orbit is called singular, if some of its point are fixed by a nontrivial element
of the Weyl group (in other words, if some elements of the orbit belong to walls of
fundamental domains of the Weyl group). Affine orbits can be divided into regular
and singular types. The regular ones are those for which the elements of the form
λ + δ form a regular orbit. Otherwise, the orbit is of singular type.
105
5.2.2
Invariants for the Cauchy–Fueter operator
Denote by ∂q̄ the Fueter operator for quaternionic valued functions in one variable q.
Consider now the operator D0 on the space of quaternionic functions f of n quaternionic variables (q1 , . . . , qn ) ∈ Hn given by D0 f = (∂q̄1 f, . . . , ∂q̄n f ).
To compare it with invariant operators on the projective quaternionic space, we
will identify the quaternionic representation H of sl1 (H) with the complex representation C2 of the complexification sl2 (C). Similarly, we will identify the quaternionic
representation Hn of sln (H) with the complex representation C2n of the complexification sl2n (C). The first one corresponds to the highest weight (1, w, 0, . . . , 0), the
other one to the highest weight (0, w0 , 1, 0, . . . , 0). There can be an invariant operator
between maps into such spaces only if these weights are at the same affine orbit of
the Weyl group. This forces the weights (and corresponding representations) to be
λ0 := (1, −2, 0, . . . , 0) ' C2 [−2], resp. λ1 := (0, −3, 1, 0, . . . , 0) ' C2n [−3]. We know
that the operator D0 should be of first order. It is possible to check directly that
the construction of first order invariant operators described above defines indeed the
operator D0 .
In more details, if f has values in the module Eλ1 , then its differential has values
in Eλ0 ⊗ Eµ , with µ = (1, −2, 1, 0 . . . , 0). The tensor product decomposes as
Eλ0 ⊗ Eµ = Eλ11 ⊕ Eλ12 ,
with λ11 = (2, −4, 1, 0, . . . , 0) and λ12 = (0, −3, 1, 0, . . . , 0). The projection to the
second summand leads to the invariant operator D0 .
Now, it is possible to write down the whole affine orbit starting with the weight
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λ0 . The list of points on the orbit is
λ2 = (0, −4, 0, 0, 1, 0, . . . , 0),
λ3 = (1, −5, 0, 0, 0, 1, 0, . . . , 0).
The general member of the list is
λj = (j − 2, −j − 2, 0, . . . , 0, 1, 0, . . . , 0)
with the 1 on the (j + 3)-th place for j = 2, . . . , 2n − 3. The last two weights of the
orbit are then
λ2n−4 = (2n − 6, −2n + 2, 0, . . . , 0, 1),
λ2n−3 = (2n − 5, −2n + 1, 0, . . . , 0).
The corresponding modules are
Eλ2 ' Λ3 (C2n )[−4],
Eλ3 ' C2 ⊗ Λ4 (C2n )[−5]; . . . ;
Eλj ' ¯j−2 (C2 ) ⊗ Λj+1 (C2n )[−j − 2]; . . . ;
Eλ2n−1 ' ¯2n−3 (C2 ) ⊗ Λ2n (C2n )[−2n − 1].
5.3
The construction of the sequence using invariant operators
In this section we show see how to apply the previous tools to construct a resolution
for the Cauchy-Fueter operator in several quaternionic variables. First we need to
107
recall some notation from Chapter 2 and introduce a complex a representation of
the operators. We will make the identification H = C ⊕ jC so that we can write a
quaternion as q = u1 + ju2 where u1 = x0 + ix1 and u2 = x2 − ix3 . The algebra
of quaternions can also be represented by 2 × 2 matrices with complex entries. For
A = 0, 1, A0 = 00 , 10 we have
·
η000 η010
η100 η110
q ' ηAA0 =
¸
·
=
x0 + ix1 −x2 − ix3
x2 − ix3 x0 − ix1
¸
·
=
u1 −u¯2
u2 u¯1
¸
and the imaginary units of quaternions are represented by the Pauli matrices
·
i'
i 0
0 −i
¸
·
,
j'
0 −1
1 0
¸
·
,
k'
0 −i
−i 0
¸
,
i=
√
−1.
We define the Cauchy-Fueter operator as
∂
∂
∂
∂
∂
=
+i
+j
+k
∂ q̄
∂x0
∂x1
∂x2
∂x3
with obvious meaning of the symbols. Differentiable functions belonging to the kernel
of ∂/∂ q̄ are called regular functions. With the previous notation, the Cauchy-Fueter
operator becomes
∂
' ∇AA0 =
∂ q̄
·
∇000 ∇010
∇100 ∇110
¸
·
=
∂x0 + i∂x1 −∂x2 − i∂x3
∂x2 − i∂x3 ∂x0 − i∂x1
while the regularity condition becomes
·
∇000 ∇010
∇100 ∇110
¸·
f0 + if1 −f2 − if3
f2 − if3 f0 − if1
¸
= 0,
¸
,
108
where a function f : H −→ H is written as f = f0 + if1 + jf2 + kf3 . Then, using the
spinor reduction [25], we write it in the form
·
¸·
∇000 ∇010
∇100 ∇110
0
0
ϕ0
0
ϕ1
¸
=0
(5.2)
0
where we have set ϕ0 := f0 + if1 and ϕ1 := f2 − if3 . In a more compact way, the
two equations in (5.2) can be written as
0
∇AA0 ϕA = 0,
A = 0, 1.
Remark 5.1. The indices in the function can be written up or down according to
their variance or covariance. If we define the matrix
·
εA0 B 0 =
0 1
−1 0
¸
we get the morphism that brings down the indices and allows the use of only covariant
symbolism:
0
ϕA0 = εA0 B 0 ϕB .
0
This notation is also useful to rewrite the regularity equation ∇AA0 ϕA = 0 in another
equivalent way as
∇A
[A0 ϕB 0 ]
=0
where the symbol [·, ·] denotes the anti-symmetrization of the two Roman indices, i.e.
∇A
[A0 ϕB 0 ]
= ∇A
A0 ϕB 0
− ∇A
B 0 ϕA0 .
This last equation allows direct computations by
making symmetrization and anti-symmetrization with respect to the Roman indices.
109
Remark 5.2. Through the rest of the chapter we will use two different types of
`
0
notation according to our needs. The symbol ηAA
0 , A, A = 0, 1,
` = 1, . . . , n is
often written as ηαA0 with A = 0, 1 α = 1, . . . , 2n, α denoting the couple of indices
(`, A). The same notation applies to the operators ∇`AA0 that become ∇αA0 . We point
out that Roman capital letters always vary within {0, 1}, small Italic letters vary
between 1 and n, while Greek letters range over 1 and 2n.
Now we will describe the operators Dj , j = 0, 1, . . . , 2n − 1 in the resulting se0
quence. Elements of the representation C2 will be denoted by ϕA , A0 = 0, 1. El0
0
ements of the symmetric power ¯j (C2 ) are symmetric tensor fields ϕA ...E with j
capital Roman indices. Elements of the outer power Λk (C2n ), k = 1, . . . , 2n − 1
are antisymmetric tensor fields ϕα,...,γ with k Greek indices. As before, the symbol
∇A0 α ; A0 = 0, 1; α = 1, . . . , 2n stands for the gradient.
The operator D0 from functions with values in Eλ0 to functions with values Eλ1
can be written as
0
0
[D0 ϕA ]α = ∇A0 α ϕA .
The operator D1 is defined by
0
[D1 ϕγ ]αβγ = ∇A0 [α ∇A
β ϕγ] ,
where the brackets [. . . ] mean anti-symmetrization in the corresponding indices. Note
that this is a second order operator.
All other operators are of first order. The operator Dj is defined on fields with
j − 2 upper indices and j + 1 lower indices by
0
0
0
0
(A0
B 0 ... F 0 )
... F A ... F
[Dj ϕB
= ∇[α ϕβ... δ] ,
β... δ ]α... δ
(5.3)
110
where the round parentheses (. . . ) mean the symmetrization in the corresponding
indices. We obtain the complex
D
D
D
1
2
0
Λ3 (C2n ) −→
C2 ⊗ Λ4 (C2n ) → . . . → ¯2n−3 (C2 ) ⊗ Λ2n (C2n ) → 0
0 → C2 −→
C2n −→
(5.4)
This sequence will be rewritten to define a complex of maps at the algebraic level
in the next section. For an explicit description in the case of two and three operators
(this latter case will recover the general procedure for n operators), see section 5.5.
5.4
Equivalence with the algebraic resolution
In the language of invariant operator theory, we can describe the Cauchy-Fueter
complex starting with the 2n × 2 matrix associated to n Cauchy-Fueter operators:


· 00 ¸
V1 (D)
 . . .  εA0 B 0 ϕ10 = Q(D)~
ϕ = 0,
ϕ
Vn (D)
where
·
Vt (D) =
∂xt0 + i∂xt1 −∂xt2 − i∂xt3
∂xt2 − i∂xt3 ∂xt0 − i∂xt1
¸
and



Q(D) = 

..
..
.
.
−∂xt2 − i∂xt3 −∂xt0 − i∂xt1
∂xt0 − i∂xt1 −∂xt2 + i∂xt3
..
..
.
.



.

111
At the level of symbols we get the matrix Q, with entries in R, that we can write in
a more compact way as
 .
..
..
.
 t
t
−η00
 η
Q =  01
t
t
 η11 −η10
..
..
.
.



.

The resolution we get is the analogue to the one described in Theorem 2.7: in this
language the only difference is that all the maps need to be translated into complex
relations and all the Betti numbers are divided by two, i.e., r00 = 2, r10 = 2n and:
µ
r`0
¶
2n − 1 n(` − 1)
=2
.
`+1
`
In particular, an analogue of (2.13) can be constructed:
Theorem 5.1. Let U be a convex open (or convex compact) set in R2n = Cn and let
S be the sheaf of infinitely differentiable functions. The complex described in Theorem
2.7 induces the exact complex
Q(D)
0
0
S 2 (U ) −→ S 2n (U ) −→ . . . −→ S r2n−2 (U ) −→ S r2n−1 (U ) −→ 0,
(5.5)
starting with the operator Q(D), which is given by the Cauchy-Fueter operators in all
the n variables.
We now wish to show that the complex obtained according to the invariant operator theory and the one computed using algebraic tools are the same, thus giving a
positive answer to the questions posed in [66] and [67] about the comparison between
the two approaches. Let us begin by recalling the following well known result (see for
112
example [54]):
Proposition 5.1. Any two minimal graded resolutions
ϕ1
. . . −→ F1 −→ F0 −→ M → 0
and
ϕ1
. . . −→ G1 −→ G0 −→ M → 0
of M are isomorphic as complexes, i.e. there are graded isomorphisms αj : Fj −→ Gj
such that αj−1 ϕj = ϕj αj , for all j ≥ 1.
Theorem 5.2. The two complexes (5.4) and (5.5) are isomorphic.
Proof. Let us consider the complex C which is the Fourier transform of the dual of
the complex (5.4). It is a complex in which the first map coincides with the map Qt ,
where Q is the Fourier transform of Q(D). The minimal free resolution of Qt has
the same length, Betti numbers and degrees of the maps as C (see Theorem 3.1 in
[66]). Now, the fact that the matrices associated to the maps appearing in C have
homogeneous entries of degree two at the first step and one in the next steps assures
that the relations they represent are not redundant. The sufficiency of these relations
is guaranteed by the fact that their number equals the number of relations found in
the minimal free resolution. So C is not only a complex but, by Proposition 5.1, it is
a minimal free resolution of Qt . This proves the statement.
113
5.5
5.5.1
Explicit compatibility conditions
The complex for 2 operators
In this section we will explicitly show how the Cauchy-Fueter complex in two quaternionic variables can be treated with the two different approaches and we show how
to translate the two descriptions into one another.
Theorem 5.3. The Cauchy-Fueter complex for two operators constructed through
the Hilbert syzygy theorem coincides with the complex constructed through invariant
operators theory.
Proof. Let us consider two Cauchy-Fueter operators and the corresponding system:
½
∂q̄1 f = g1
∂q̄2 f = g2 .
This system can be translated into another system of eight real equations that can
be written in matrix form (see Section 3) as
P (D)f = g.
Recall that the minimal free resolution of the module M is given by 2.19:
Pt
Pt
Pt
2
1
0 −→ R4 −→
R8 −→
R8 −→ R4 −→ M −→ 0
which translates into the sheaf complex
P (D)
P1 (D)
P2 (D)
0 −→ S(U )4 −→ S(U )8 −→ S(U )8 −→ S(U )4 −→ 0.
We know from the general theory of the Cauchy–Fueter complex [2–4], that the two
114
quaternionic relations coming from the matrix P1 (D) give the following quaternionic
compatibility conditions:
½
∂q̄1 ∂q1 g2 − ∂q̄2 ∂q1 g1 = 0
∂q̄2 ∂q2 g1 − ∂q̄1 ∂q2 g2 = 0.
(5.6)
The complex closes with one more linear condition that is the compatibility condition
for the solvability of the system
½
∂q̄1 ∂q1 g2 − ∂q̄2 ∂q1 g1 = h12
∂q̄2 ∂q2 g1 − ∂q̄1 ∂q2 g2 = h21 .
(5.7)
One can easily verify that the condition, coming from P2 (D), is
∂q1 h21 + ∂q2 h12 = 0.
(5.8)
Now we consider the description arising from invariant theory. We define the usual
Cauchy-Fueter operators (and their conjugates) as
·
∂q̄i ∼
= ∇iAA0 =
∇i000 ∇i010
∇i100 ∇i110
¸
·
,
¯i 0 =
∂qi ∼
=∇
AA
∇i110 −∇i010
−∇i100
∇i000
¸
.
(5.9)
Taking into account the above definitions we have that
·
∂q̄i ∂qj ∼
=
∇i000 ∇j110 − ∇j100 ∇i010
∇i100 ∇j110 − ∇i110 ∇j100
−∇i000 ∇j010 + ∇i010 ∇j000
−∇i100 ∇j010 + ∇i110 ∇j000
¸
.
In particular with i = j we have the Laplace operators
·
∂q̄i ∂qi = ∂qi ∂q̄i ∼
=
∇i000 ∇i110 − ∇i010 ∇i100
0
0
∇i110 ∇i000 − ∇i100 ∇i010
¸
.
115
According to the previous section, the complex in the case of two operators can be
described as follows:
D
D
D
0
1
2
0 −→ C2 −→
C4 −→
Λ3 (C4 ) −→
C2 ⊗ Λ4 (C4 ) −→ 0.
The compatibility relations for the Cauchy–Fueter system are
∇iA
[A0 ϕB 0 ]
= ψAi ,
i = 1, 2, A, B ∈ {0, 1},
taking the anti-symmetrization and symmetrization according to (5.3), it is possible
to show that the compatibility conditions are given by the 3 × 3 minors of the matrix

∇1000 ∇1010 ψ01



 1

1
1
 ∇100 ∇110 ψ1 



.
 2

 ∇ 0 ∇2 0 ψ02 
00
01




2
2
2
∇100 ∇110 ψ1
In fact, consider the four different minors
¯ i
¯ ∇000 ∇i010 ψ0i
¯
¯
¯
ij
MA = ¯¯ ∇i100 ∇i110 ψ1i
¯
¯ j
¯ ∇ 0 ∇j 0 ψ j
A
A0
A1
¯
¯
¯
¯
¯
¯
¯
¯
¯
¯
i, j ∈ {1, 2}, A ∈ {0, 1}
and observe that the relations
M012 = 0,
M112 = 0
116
can be written as

·
M012
M112
¸

=
∇1000 ∇2110 − ∇1010 ∇2100
−∇1000 ∇2100 + ∇1010 ∇2000 ∆1
∇1100 ∇2110 − ∇1110 ∇2100
−∇1100 ∇2010 + ∇1110 ∇2000
0

1
ψ
0

 ψ11 
0




 2  = 0.
∆ 1  ψ0 
ψ12
(5.10)
By setting gi = [ψ0i ψ1i ]t , one verifies that condition (5.10) corresponds to
£
−∂q̄2 ∂q1 ∂q̄1 ∂q1
¤
·
g1
g2
¸
= 0,
and, analogously, the relations
M021 = 0,
M121 = 0.
(5.11)
correspond to ∂q̄2 ∂q2 g1 − ∂q̄1 ∂q2 g2 = 0.
Let us consider the inhomogeneous system arising from (5.10), (5.11):

M012 = ϕ10




 M112 = ϕ11


M021 = ϕ20


 21
M1 = ϕ21 .
To close the complex we need to consider the relations one obtains by suitable symmetrization and anti-symmetrization of the indices. Once again, it is equivalent to
117
consider the determinants of the two 4 × 4 matrices

∇10A0 ∇1000 ∇1010 ψ01



 1

1
1
1
 ∇1A0 ∇100 ∇110 ψ1 




 2

2
2
2
 ∇ 0 ∇ 0 ∇ 0 ψ0 
00
01
 0A



∇21A0 ∇2100 ∇2110 ψ12
A0 ∈ {0, 1}.
The determinants, for A0 = 0, 1, give the two conditions:
∇1000 ϕ21 − ∇1100 ϕ20 + ∇2000 ϕ11 − ∇2100 ϕ10 = 0,
∇1010 ϕ21 − ∇1110 ϕ20 + ∇2010 ϕ11 − ∇2110 ϕ10 = 0,
that, in matrix form, can be written in the form

·
∇2110 −∇2010
∇1110 −∇1010
2
2
−∇100
∇000 −∇1100
∇1000

ϕ10
¸  ϕ11 




 2  = 0.
 ϕ0 
ϕ21
(5.12)
Using (5.9) and setting h12 = [ϕ10 ϕ11 ], h21 = [ϕ20 ϕ21 ], it is immediate to verify that
(5.12) corresponds to the relation (5.8) in the “algebraic” complex.
The description of the maps in the complex of two Cauchy-Fueter operators is not
new, see e.g. [63]. The relations involved come from the so called radial algebra:
Definition 5.2. Let S be a set of objects {x, y, z, . . .}. The radial algebra R(S) is
defined to be the associative algebra generated by S over R with the defining relations
[{x, y}, z] = xyz + yxz − zxy − zyx = 0, for x, y, z ∈ S.
(5.13)
118
Note that the radial algebra relations (5.13) alone are not useful to find the all
the syzygies in the case of more than two Cauchy-Fueter operators, as only some of
them can be written in this way. It has been shown in [63] that the radial algebra
allows to write all the maps in the complex of three Dirac operators with values in a
Clifford algebra over at least five imaginary units.
Remark 5.3. In the case of the algebraic construction, the fact that the relations
arising from the radial algebra are not only necessary but also sufficient was proved
with the use of CoCoA which provides the minimal number of relations at each step.
However CoCoA (and similar packages, like Macaulay2 or Singular) cannot display
the relations in quaternionic form since the syzygies are written in real form and, in
general, it is not possible to automatically group the various real relations to obtain
quaternionic ones. The main advantage of the construction through the representation
theory is that it provides a method to write explicit complex relations which take into
account the invariance of the operators involved, so they allow to be easily grouped
to produce quaternionic relations.
We conclude this section presenting a duality theorem that describes the space of
hyperfunctions in 2 quaternionic variables through the last map P2 (D) of the Cauchy–
Fueter complex (see also [25, 50]). Since this map is associated to the conjugate of
the Cauchy-Fueter operator, we obtain a characterization of the dual of the space of
hyperfunctions in terms of the kernel of D2 .
Theorem 5.4. Let K be a compact convex set in H2 and let Q = D2 be the matrix
symbol of the conjugates of two Cauchy–Fueter operators ∂q1 and ∂q2 . Let S and Rl
be the sheaf of infinitely differentiable functions and of (left) regular functions in H2 ,
119
respectively. Then
t
3
HK
(H2 , S Q ) ∼
= [Rl (K)]0
(5.14)
and
t
3
HK
(H2 , Rl ) ∼
= [S Q (K)]0 .
For a more detailed treatment of the topic of hyperfunctions in two quaternionic
variables see [23, 24].
5.5.2
The complex for n ≥ 3 operators
In the case of more than 2 Cauchy-Fueter operators we know the length of the complex, the number of relations at each step and their degree from Theorems 2.6 and
2.7. However an explicit description of the maps appearing in the complex has not
been given yet (with the only exception of the first syzygies). The presence of the
exceptional syzygies (see section 2.3) which involves operators containing only two of
the four real derivatives, makes it hard (and perhaps impossible) to write the next relations using only the Cauchy–Fueter operators in the various quaternionic variables.
The procedure illustrated in this chapter allows to provide the needed description of
all the maps. The first syzygies appearing in the complex are as in theorem 2.6.
Theorem 5.5. The first compatibility conditions for the Cauchy-Fueter system in
three quaternionic variables can be obtained via the complex (5.4).
Proof. According to the notations introduced in Section 5.3, elements of the represen0
tation C2 will be denoted by ϕA , A0 = 0, 1. Elements of the symmetric power ¯j (C2 )
0
0
are symmetric tensor field ϕA ...E with j capital Roman indices. Elements of the
outer power Λk (C2n ), k = 1, . . . , 2n − 1 are anti-symmetric tensor fields ϕα,...,γ with
120
k Greek indices. The symbol ∇A0 α , A0 = 0, 1; α = 1, . . . , 2n denotes the gradient.
Let us consider first the case n = 3. The complex arising from this construction is
D
D
D
D
D
0 → C2 →0 C6 →1 Λ3 (C6 ) →2 C2 ⊗Λ4 (C6 ) →3 ¯2 (C2 )⊗Λ5 (C6 ) →4 ¯3 (C2 )⊗Λ6 (C6 ) −→ 0.
(5.15)
From the previous discussion, one may argue that the 3 × 3 minors of the matrix


















∇1000 ∇1010 ψ01
∇1100
∇1110
∇2000 ∇2010
∇2100
∇2110
∇3000 ∇3010
∇3100 ∇3110







ψ02 



2 
ψ1 


3 
ψ0 

3
ψ1
ψ11
(5.16)
give the compatibility relations for the system
∇iA [A0 ϕB 0 ] = ψAi ,
i = 1, 2, 3,
A, B, A0 , B 0 ∈ {0, 1}
to have a solution.
We define the following 12 minors for A = 0, 1; i, j = 1, 2, 3:
¯ i
¯ ∇000 ∇i010 ψ0i
¯
¯
¯
ij
MA = ¯¯ ∇i100 ∇i110 ψ1i
¯
¯ j
¯ ∇ 0 ∇j 0 ψ j
A
A0
A1
¯
¯
¯
¯
¯
¯
¯
¯
¯
¯
121
and also the 8 minors for A, B, C = 0, 1:
MABC
¯ 1
¯ ∇A00 ∇1A10 ψA1
¯
¯
¯
= ¯¯ ∇2B00 ∇2B10 ψB2
¯
¯ 3
¯ ∇ 0 ∇3 0 ψ 3
C
C1
C0
¯
¯
¯
¯
¯
¯
¯
¯
¯
¯
It is now possible to show that the minors MAij correspond to the 6 quaternionic
syzygies involving just two indices i, j ∈ {1, 2, 3}, i.e.
∂q̄i ∂qi gj − ∂q̄j ∂qi gi = 0.
(5.17)
For sake of simplicity we work with i = 1 and j = 2, since the other cases are similar.
The relation MA12 = 0, A = 0, 1 can be written as





∇1000 ∇2110 − ∇1010 ∇2100 −∇1000 ∇2100 + ∇1100 ∇2000 ∆1
∇1010 ∇2110 − ∇1110 ∇2010 −∇1010 ∇2100 + ∇1110 ∇2000
0
0
∆1

ψ01
 1 
  ψ1 

0 0 
 2 
  ψ0 
  2  = 0 (5.18)
  ψ1 



0 0 
 3 
 ψ0 
ψ13
where we have set ∆1 = ∇1000 ∇1110 − ∇1010 ∇1100 . We now observe that the equation
(5.18) correspond to (5.17) for i = 1, j = 2. Similarly, the other syzygies involving
only two indices i, j can be obtained as
M0ij = M1ij = 0.
Let us consider the syzygies of the form (2) in theorem 2.6. The triples of indices
122
(h, r, s) take the values 1 ≤ h, r, s ≤ 3 so that we get two independent relations, e.g.
∂q1 ∂q̄2 g3 + ∂q2 ∂q̄1 g3 − ∂q̄3 ∂q1 g2 − ∂q̄3 ∂q2 g1 = 0
(5.19)
∂q1 ∂q̄3 g2 + ∂q3 ∂q̄1 g2 − ∂q̄2 ∂q3 g1 − ∂q̄2 ∂q3 g1 = 0.
(5.20)
and
We may verify that the syzygies above correspond to the following systems:
½
½
M010 − M100 = 0
M011 − M101 = 0
and
M110 − M011 = 0
M100 − M001 = 0
(5.21)
in fact, the system on the left in (5.21) can be written in matrix form as

£
¯2 0
−∇3AA0 ∇
AA
¯1 0
−∇3AA0 ∇
AA
¯ 1 0 ∇2 0 + ∇
¯ 2 0 ∇1 0
∇
AA
AA
AA
AA

ψ01
 ψ11 






¤  ψ02 
 2  = 0.
 ψ1 




 3 
 ψ0 
ψ13
By setting gi = [ψ0i ψ1i ]t , it is easy to verify that this matrix relation corresponds to
(5.19). In an analogous way, the system on the right corresponds to (5.20).
Finally Let us consider the syzygies of type (3). The operators Dqi , Dq0 i , Dq00i can
be represented by the following 2 × 2 matrices with complex entries:
·
Dqi ∼
=
0 ∇i010
∇i100
0
Dq00i
¸
1
∼
=
2
Dq0 i
·
1
∼
=
2
·
∇i110 − ∇i000 ∇i100 + ∇i010
∇i010 + ∇i100 ∇i000 − ∇i110
∇i000 − ∇i110 ∇i100 − ∇i010
∇i010 − ∇i100 ∇i110 − ∇i000
¸
¸
123
then the compatibility relations can be expresses as
(Dq2 ∂q̄3 − Dq3 ∂q̄2 )g1 + (Dq3 ∂q̄1 − Dq1 ∂q̄3 )g2 + (Dq1 ∂q̄2 − Dq2 ∂q̄1 )g3 = 0,
(5.22)
(Dq0 2 ∂q̄3 − Dq0 3 ∂q̄2 )g1 + (Dq0 3 ∂q̄1 − Dq0 1 ∂q̄3 )g2 + (Dq0 1 ∂q̄2 − Dq0 2 ∂q̄1 )g3 = 0.
(5.23)
With some computations similar to those already done in the other cases, one gets
that (5.22) and (5.23) correspond respectively to the systems:
½
M101 − M110 = 0
M001 − M010 = 0
½
and
M000 = 0
M111 = 0.
(5.24)
Remark 5.4. Note that this description is not affected by the fact we are considering
only three variables, since it can be repeated for any choice of indices A, B, C in the
definition of the matrices MAij and MABC . So the proof holds in the general case
n ≥ 3.
Remark 5.5. In the last section we will see how to perform computations with the
new symbols η using a computer. We just want to highlight now that the use of the
CoCoA command GenRepr has been particularly useful to find the relations between
the quaternionic syzygies and minors defined above.
We can go further with the description of the maps in the complex. Let us consider
again the case of three operators. We consider the nonhomogeneous system

∂q̄r ∂qs gs − ∂q̄s ∂qs gr = hrs
r, s = 1, 2, 3




 ∂q1 ∂q̄2 g3 + ∂q2 ∂q̄1 g3 − ∂q̄3 ∂q2 g1 − ∂q̄3 ∂q1 g2 = a1
∂q3 ∂q̄1 g2 + ∂q1 ∂q̄3 g2 − ∂q̄2 ∂q1 g3 − ∂q̄2 ∂q3 g1 = a2


(Dq1 ∂q̄2 − Dq2 ∂q̄1 )g3 + (Dq2 ∂q̄3 − Dq3 ∂q̄2 )g1 + (Dq3 ∂q̄1 − Dq1 ∂q̄3 )g2 = b1 ,



(Dq0 1 ∂q̄2 − Dq0 2 ∂q̄1 )g3 + (Dq0 2 ∂q̄3 − Dq0 3 ∂q̄2 )g1 + (Dq0 3 ∂q̄1 − Dq0 1 ∂q̄3 )g2 = b2 .
124
The second syzygies can be obtained as the maximal minors of the matrix obtained
by adding to (5.16) a column of the type








∇10A0
∇11A0
∇20A0
∇21A0
∇30A0
∇31A0








(5.25)
where A0 = 0, 1. This amounts to compute ∇A
[α Mβγδ] where A = 0, 1 and α, . . . , δ
are different indices associated to the rows β, γ, δ of the matrix and thus varying in
{1, . . . , 6}. When dealing with 4×4 minors involving only two different upper indices,
we get relations of the type (5.12) which can be rewritten as
∂qi hji + ∂qj hij = 0.
Remark 5.6. Although we do not write explicitly here all the second syzygies, we
wish to point out that not all the relations can be written using the Cauchy-Fueter
operator. For example, again in the case n = 3, we obtain
1
1
1
Dq3 h21 − Dq2 h31 − Dq1 a1 + Dq1 a2 − Ďq1 b1 − Ďq1 B2 = 0
3
3
3
Ďq3 h21 − Ďq2 h31 −
1
1
1
Ďq1 a1 + Ďq1 a2 − Ďq1 b1 − Ďq1 B2 = 0,
3
3
3
where we have set Ďqi = (∂x0i + i∂x1i ).
Following the same procedure, we can write not only the third syzygies by computing
the minors of the 5 × 6 matrices that we obtain by adding two columns of the type
(5.25) to the matrix (5.16) and all the other syzygies in the resolution. In the following
125
t
theorem we describe the map P2n−2
(D), last in the complex.
t
Theorem 5.6. The last map P2n−2
(D) in the Cauchy-Fueter complex in n ≥ 3
operators is given by





∂q1 .. ∂qn 0
0 .. 0 Ďq1
..
.
..
.
. .. ..
0 .. 0
0
.. 0
0
.. Ďqn −Dq1
..
.
.
.. ..
.. 0
0
..
0
.. 0
.. −Dqn .. 0
.
..
.
.. ..
..
..
0
.. Ďq1
..
..
..
..
0
0
..
.
0
0
..
.
Ďqn −Dq1
..
..
0
0
..
.
..
.. −Dqn



.

(5.26)
Proof. At the final step we have to consider the matrices obtained from



















∇1000 ∇1010 ψ01
∇1100
∇1110
ψ11
∇2000 ∇2010 ψ02
∇2100 ∇2110 ψ12
..
..
..
.
.
.
∇n000 ∇n010 ψ0n



















(5.27)
∇n100 ∇n110 ψ1n
by adding 2n − 3 columns of the type

∇10A0
∇11A0
..
.




 n
 ∇0A0
∇n1A0







where A0 = 0, 1. The index A runs on the 2n−2 possibilities: (0, 0, . . . , 0), (1, 0, . . . , 0),
(1, 1, . . . , 0), (1, 1, . . . , 1). We obtain 2n − 2 square matrices of dimension 2n whose
126
determinants can be written, according to the Laplace theorem, by multiplying each
elements in the leftmost column by the corresponding minors. Note that only the
first two possibilities (0, 0, . . . , 0), (1, 0, . . . , 0) involve the same (2n − 1) × (2n − 1)
minors. The matrix representing Q can be written as







∇1110 .. −∇n010
−∇1100 .. ∇n000
..
..
.
..
.
0
..
0
0
..
0
..
..
0
0
..
.
0
0
..
.
..
..
0
0
..
.
0
0
..
.
0
0
..
.
0
0
..
.
..
..
0
0
..
.
0
0
..
.
..
..
..
1
n
1
.. ∇000
0 .. ∇000
0
0
−∇100 ..
0
−∇1100
.. 0 ∇1110 .. 0 ∇n110 −∇1010
0
.. −∇n010
0
(5.28)
or, in quaternionic form as (5.26).
Last map in the complex is the most important in our description since it allows to
prove a duality theorem which is related to the definition of hyperfunctions in several
quaternionic variables. Let us recall the following theorem (see [50]):
t
Theorem 5.7. Let K be a compact convex set in Hn and set Q = P2n−2
. Let S and
Rl be the sheaf of infinitely differentiable functions and of (left) regular functions,
respectively. Then
t
2n−1
HK
(Hn , S Q ) ∼
= [Rl (K)]0
(5.29)
and
t
2n−1
HK
(Hn , Rl ) ∼
= [S Q (K)]0 .
t
The sheaf S Q is described as follows
t
Proposition 5.2. The elements of the sheaf S Q are (n−1)-tuples F = (f1 , . . . , fn−1 )t
of infinitely differentiable functions such that f1 is regular with respect to the variables







127
q1 , . . . , qn , while fj , j ≤ 2 are anti-regular in the variables q1 , . . . , qn and satisfy
(∂x0` − i∂x1` )fj = 0 for any ` = 1, . . . , n.
Proof. Let us consider the system P2n−2 (D)F = 0 where P2n−2 (D) is given is (5.26).
This translates into the fact that f1 is regular with respect to all the variables q1 , . . . , qn
while each fj , j = 2, . . . n−1 satisfy the system (∂x0` −i∂x1` )fj = (j∂x2` +k∂x3` )fj = 0,
for any ` = 1, . . . n. By taking the difference of these last two relations we get the
statement.
5.6
Some computations with CoCoA
The statement of theorem 5.2 has been confirmed by the computations carried out in
the previous section for 2 and 3 Cauchy-Fueter operators. The resolutions we obtain
with the use of invariant operators is the same that the algebraic analysis offers in
those cases. As an additional support to this result we may think of performing the
calculation of the compatibility conditions of the Cauchy-Fueter system in 2 and 3
quaternionic variables using the complex representation (5.2). In other words, instead of writing the usual Fourier-Laplace transform of the matrix P (D) using real
variables, we can cut the dimension of the problem by a factor of two if we make use
of the symbols ∇jAA0 .
Case n = 2. We will work in the ring
R := C[a1 , . . . , a4 , b1 , . . . , b4 ]
128
and make the following identification:
∇1000 → a1 ,
∇1010 → a2 ,
∇1100 → a3 ,
∇1110 → a3 ,
∇2000 → b1 ,
∇2010 → b2 ,
∇2100 → b3 ,
∇2110 → b3 .
With this notation the matrix symbol of the Cauchy-Fueter operator in 2 quaternionic
variables becomes

a1
 a3
C2 := 
 b1
b3

a2
a4 
.
b2 
b4
(5.30)
CoCoA can be used to define such matrix and calculate the free resolution of its cokernel M2 := R2 /hC2t i as follows:
Use R::=Q[a[1..4]b[1..4]];
A2:=Coala.ComplexMat(Indets());
M2:=Module(A2);
Res(M2);
0 --> R^2(-4) --> R^4(-3) --> R^4(-1)
------------------------------As we can see it is exactly equivalent, modulo a factor of 2 in the Betti numbers, to
2.19. it is possible also to check that the maps we obtain in this resolution are the
same as described in section 5.5.1:
ResM2:=MyResMap(M2);
Mat(ResM2[2]); -- first syzygies
Mat([[-a[4]b[1]+a[3]b[2], a[2]b[1]-a[1]b[2], -a[2]a[3]+a[1]a[4], 0],
129
[-a[4]b[3]+a[3]b[4], a[2]b[3]-a[1]b[4], 0, -a[2]a[3]+a[1]a[4]],
[-b[2]b[3]+b[1]b[4], 0, a[2]b[3]-a[1]b[4], -a[2]b[1]+a[1]b[2]],
[0, -b[2]b[3]+b[1]b[4], a[4]b[3]-a[3]b[4], -a[4]b[1]+a[3]b[2]]])
------------------------------Mat(ResM2[3]); -- second syzygies, last map
Mat([
[b[3], -b[1], a[3], -a[1]],
[-b[4], b[2], -a[4], a[2]]
])
------------------------------Evidently the first syzygy map corresponds (modulo a sign in some of the rows) to
the system given by the equations in 5.10 and 5.11, while the last map correspond to
the system 5.12.
Case n = 3 The same procedure can be adapted to the case of three quaternionic
variables. With obvious meaning of symbols, we define the matrix




C3 := 



a1
a3
b1
b3
c1
c3
a2
a4
b2
b4
c2
c4




.



in the ring of complex coordinates R := C[a1 , . . . , c4 ]. The resolution of the module
M3 := R2 /hC3t i is then computed with CoCoA as follows
Use R::=Q[a[1..4]b[1..4]c[1..4]];
130
A3:=Coala.ComplexMat(Indets());
M3:=Module(A3);
Res(M3);
0 --> R^4(-6) --> R^18(-5) --> R^30(-4) --> R^20(-3) --> R^6(-1)
------------------------------The maps arising in the free resolution correspond again to the ones described in
section 5.5.2. I will show here only the first syzygies and the last map. The system
associated to the first map correspond to the equations given by the minors of the
matrix (5.16). By grouping the rows in pairs, it is possible to recognize the syzygies
2.6 in quaternionic form.
ResM3:=Coala.MyResMap(M3);
Mat(ResM3[1]);
Mat([
[-a[4]b[1]+a[3]b[2], a[2]b[1]-a[1]b[2], -a[2]a[3]+a[1]a[4], 0, 0, 0],
[-a[4]b[3]+a[3]b[4], a[2]b[3]-a[1]b[4], 0, -a[2]a[3]+a[1]a[4], 0, 0],
[-b[2]b[3]+b[1]b[4], 0, a[2]b[3]-a[1]b[4], -a[2]b[1]+a[1]b[2], 0, 0],
[0, -b[2]b[3]+b[1]b[4], a[4]b[3]-a[3]b[4], -a[4]b[1]+a[3]b[2], 0, 0],
[-a[4]c[1]+a[3]c[2], a[2]c[1]-a[1]c[2], 0, 0, -a[2]a[3]+a[1]a[4], 0],
[-b[2]c[1]+b[1]c[2], 0, a[2]c[1]-a[1]c[2], 0, -a[2]b[1]+a[1]b[2], 0],
[0, -b[2]c[1]+b[1]c[2], a[4]c[1]-a[3]c[2], 0, -a[4]b[1]+a[3]b[2], 0],
[-b[4]c[1]+b[3]c[2], 0, 0, a[2]c[1]-a[1]c[2], -a[2]b[3]+a[1]b[4], 0],
[0, -b[4]c[1]+b[3]c[2], 0, a[4]c[1]-a[3]c[2], -a[4]b[3]+a[3]b[4], 0],
[0, 0, -b[4]c[1]+b[3]c[2], b[2]c[1]-b[1]c[2], -b[2]b[3]+b[1]b[4], 0],
[-a[4]c[3]+a[3]c[4], a[2]c[3]-a[1]c[4], 0, 0, 0, -a[2]a[3]+a[1]a[4]],
131
[-b[2]c[3]+b[1]c[4], 0, a[2]c[3]-a[1]c[4], 0, 0, -a[2]b[1]+a[1]b[2]],
[0, -b[2]c[3]+b[1]c[4], a[4]c[3]-a[3]c[4], 0, 0, -a[4]b[1]+a[3]b[2]],
[-b[4]c[3]+b[3]c[4], 0, 0, a[2]c[3]-a[1]c[4], 0, -a[2]b[3]+a[1]b[4]],
[0, -b[4]c[3]+b[3]c[4], 0, a[4]c[3]-a[3]c[4], 0, -a[4]b[3]+a[3]b[4]],
[0, 0, -b[4]c[3]+b[3]c[4], b[2]c[3]-b[1]c[4], 0, -b[2]b[3]+b[1]b[4]],
[-c[2]c[3]+c[1]c[4], 0, 0, 0, a[2]c[3]-a[1]c[4], -a[2]c[1]+a[1]c[2]],
[0, -c[2]c[3]+c[1]c[4], 0, 0, a[4]c[3]-a[3]c[4], -a[4]c[1]+a[3]c[2]],
[0, 0, -c[2]c[3]+c[1]c[4], 0, b[2]c[3]-b[1]c[4], -b[2]c[1]+b[1]c[2]],
[0, 0, 0, -c[2]c[3]+c[1]c[4], b[4]c[3]-b[3]c[4], -b[4]c[1]+b[3]c[2]]])
------------------------------The last map that closes the complex corresponds to the one described in theorem 5.6,
in particular it is equivalent to (5.28) modulo signs and elementary column operations.
Here is its transpose:
















t
Q =
















−a4
a3
0
0
a2 −a1
0
0 

−b4
b3
0
0 

b2 −b1
0
0 

−c4
c3
0
0 

c2 −c1
0
0 

−a1
0
a2
0 

a3
0 −a4
0 

−b1
0
b2
0 

b3
0 −b4
0 

−c1
0
c2
0 

c3
0 −c4
0 

0
0 −a1
a2 

0
0
a3 −a4 

0
0 −b1
b2 

0
0
b3 −b4 

0
0 −c1
c2 
0
0
c3 −c4
Chapter 6: Computation of Dirac syzygies using
megaforms
The syzygies presented in section 2.4 for the Dirac operator acting on Clifford-valued
functions have been calculated using both the approach described in Chapter 2 and
using the radial relations (2.27). Unfortunately, real computations with CoCoA are
often too heavy and outputs are too big to be easily read and converted into expressions containing only Clifford variables. This is why the Betti diagrams (2.4) are
still incomplete and even when a dimension of a syzygy module is known it is hard
to describe explicitly its generators in a compact way. In [62] the authors propose
a different approach, which make no use of the real expression of the Dirac operators and utilizes only the radial relations. This approach is based on the concept of
megaforms. Roughly speaking, one considers a set of ”abstract vector variables” S,
such as the set of Dirac operators ∂x1 , . . . , ∂xk and considers the associative algebra
generated by the elements of S. The proper space where to make computations is
the quotient between the tensor algebra generated by S and the two-sided ideal
I = h[{x, y}, z] | x, y, z ∈ Si.
The next step is to generalize the definition of a differential ”d =
P
∂xj dxj ” to the
case of abstract vector derivatives. Let us rename ∂xj with ∂j . Dirac operators can
play the role of derivatives in the formula for a differential, but one has to define the
n
.
right objects to play the role of the forms dxj . These are new symbols Djn and Dij
132
133
Such symbols will be associated to the corresponding first or second order derivative
2
will
with respect to the variables given by the indices. For example, the symbol D13
be associated to the second derivative ∂1 ∂3 for the differential d2 , i.e. the second
map of the complex. While in the case of forms on Rm the differentials dn consist of
a sum of degree one forms with first order derivatives as coefficients, in the case of
several Dirac operators we have to take into account that the syzygies we are looking
for could be of degree one or two, given that the radial relations are expressed by
polynomials of degree three.
A theory of megaforms can also be formulated for the Cauchy-Fueter complex as
explained in [23, 24], were we completely treat the case of regular functions in two
quaternionic variables. This approach turns out to have significative applications to
the characterization of quaternionic hyperfunctions in two variables.
6.1
The space of megaforms
Let us recall that we are dealing with k Dirac operators ∂1 , . . . , ∂k acting on differentiable functions f : (Rm )k −→ Cm . We can give a precise algebraic definition of the
objects described above. The set of basic megaforms of degree n ≥ 0 is the collection
of symbols
n
Bn := {Din , Dj`
| i, j, ` = 1, . . . , m},
and the set of basic megaforms will be the union
B :=
∞
[
n=0
Bn .
(6.1)
134
The radial algebra will be the associative algebra generated by the k Dirac operators
together with the radial relations (2.27):
R := h∂1 , . . . , ∂k | [{∂i , ∂j }, ∂` ] = 0i.
(6.2)
We will consider non commutative polynomials in B with (right) coefficients in R
to be our main algebraic setting. Some particular polynomials will play the role of
differentials:
d0 :=
d1 :=
Pk
i=1
Di0 ∂i ,
Pk
i,j=1
1
Dij
∂i ∂j ,
dn = dn1 + dn2 =
Pk
i=1
(6.3)
Din ∂i +
Pk
i,j=1
n
Dij
∂i ∂j ,
n > 1.
Definition 6.1. Let ∂1 , . . . , ∂k be Dirac operators and define B,R and dn as above.
The space Mk of free megaforms with right coefficients in R is the associative algebra
which is generated over R by the set B together with the identities derived by the
closure conditions dn+1 · dn = 0, for every n ∈ N.
Remark 6.1. It is extremely important to note that, since we will be interested in the
relation coming out of the closure conditions dn+1 dn = 0, the expression containing
megaforms will always involve sequences of basic megaforms of the type
. . . Dn+1 Dn Dn−1 . . .
for some choices of indices i, j, k. In other words, if we see a differential dn as a
non commutative polynomial Mk , the capitalized variables appear as ordered with
increasing superscripts from right to left. Because of this, it is possible to simplify the
135
notation and avoid to indicate the superscript of each basic megaform, being there
just one possible choice at each step given the initial one.
Let us introduce the concept of space of n-megaforms Fn . For n = 0 we define it
to be simply the space of differentiable functions
F0 := C ∞ ((Rm )k , Cm ).
The set of 1-megaforms will be space of elements of the type g =
Pm
i=1
Di1 gi , gi ∈ F0
and recursively we define Fn to be generated by the set Bn with lower dimensional
forms as right coefficients as follows:
Fn := hBn iFn−1 ,
n > 1.
(6.4)
With all the above definition it is immediate to interpret the differential dn as a
(multiplication) map between the spaces Fn and Fn+1 so that from their definition
and form the closure conditions we obtain a complex
d0
d1
d2
d3
d4
dN −1
0 −→ F0 −→ F1 −→ F2 −→ F3 −→ F4 −→ . . . −→ FN −→ 0.
(6.5)
Remark 6.2. As illustrated in [62], complex (6.5) is an analogue of the Dolbeault
sequence for forms on Rm . Because it is built with the only use of the radial relations
on the symbols ∂j , the complex is an analytic representation of the Dirac complex
associated to k Dirac operators as introduced in Section 2.4.
6.2
Explicit construction of the Dirac complex
The construction of the Dirac complex in the radial case using megaforms consists
of two basic steps [62]. First, one finds the image of dn in the space of megaforms
136
Fn+1 . The closure condition dn+1 dn = 0 together with the radial relations, are used
to derive suitable relations satisfied by the megaforms. We will call this part of the
construction the closure step. The second step is to use the relations just computed
to find syzygies for the non-homogeneous system dn−1 gn−1 = gn . We will denote this
step by syzygy step. We will briefly illustrate the application of this method with an
example using 3 Dirac operators. Note that in this example, we can use that fact
that d11 = 0, already proved in [62].
Example 6.1. Let d0 =
P3
i=1
Di ∂i and d1 =
P3
i,j=1
Dij ∂i ∂j . The condition d12 d0 = 0
implies that for any f ∈ F0 it is
3
X
Dki Dj ∂k ∂i ∂j f = 0.
i,j,k=1
By writing the right hand term explicitly, by using the radial algebra defining relations
and grouping the various terms, we get
Dij Di = 0 i, j ∈ {1, 2, 3}
Dii Dj + Dji Di = 0 i, j ∈ {1, 2, 3} i 6= j
Dik Dj + Djk Di = 0 (i, j, k) = (1, 3, 2), (3, 2, 1), (2, 1, 3)
D23 D1 + D31 D2 + D12 D3 = 0
(6.6)
We now study the kernel of the map d1 : F1 → F2 and we show that a 1–form
g is d1 –closed if and only if its components gj satisfy the compatibility conditions of
the system d0 f = g. Let g =
P3
i=1
Di gi be an element of F1 . By the definition of g
and d1 we have that d1 g = 0 can be written as
3
X
i,j,k=1
Dij Dk ∂i ∂j gk = 0.
137
In view of (6.6) this can be rewritten in the form
D12 D3 ({∂2 , ∂3 }g1 − ∂1 ∂2 g3 − ∂1 ∂3 g2 )+
D31 D2 ({∂1 , ∂3 }g2 − ∂2 ∂1 g3 − ∂2 ∂3 g1 )+
P3
2
i,j=1,i6=j Dii Dj (∂j ∂i gi − ∂i gj ) = 0.
Setting the coefficients of the independent symbols of the forms equal to zero, we get
∂j ∂i gi − ∂i2 gj = 0, i, j = 1, 2, 3 i 6= j
{∂2 , ∂3 }g1 − ∂1 ∂2 g3 − ∂1 ∂3 g2 = 0
{∂1 , ∂3 }g2 − ∂2 ∂1 g3 − ∂2 ∂3 g1 = 0
(6.7)
i.e. d1 g = 0 if and only if (g1 , g2 , g3 ) satisfy the compatibility conditions for the
solvability of the system

 ∂1 f = g1
∂2 f = g2

∂3 f = g 3 .
Remark 6.3. The computations which we have just concluded show that the space
F2 is generated by 8 different symbols of the type Dij Dk . More precisely a 2–form
h ∈ F2 can be written as
h = D33 D1 h331 + D22 D1 h221 + D11 D2 h112 + D11 D3 h113
+D22 D3 h223 + D33 D2 h332 + D31 D2 h312 + D12 D3 h123 .
As the example has shown, this approach has produced 9 relations on the space of
megaforms (6.6) with the first closure step and consequently 8 syzygies for the data
g1 , g2 , g3 with the first syzygy step. As forecast in [62], such procedure can be, at
least in theory, generalized and carried on iteratively until we reach the end of the
complex. However, computations of the closure and syzygy steps for the next spaces
138
of the complex happen to be extremely heavy to be performed ”by hand”. This is
why I have tried to use CoCoA to accomplish this task.
6.3
The idea on CoCoA
When analyzing carefully the objects involved in the computation of syzygies using
megaforms, we notice that to represent the differentials dn we cannot avoid the non
commutative nature of the problem. However, with CoCoA it is possible to handle
lists of any type. A list is a more general object than an array or vector, as it may
contain even entries that are not of the same type. Furthermore, in Remark 6.1 we
have noticed that the megaforms have a peculiar form when arising from the closure
2
0
conditions. For example, the monomial D13
D21 D21
is admissible for computations
involved in the construction of the complex, because the superscripts of its elements
are increasing from right to left. We could expect to obtain something like it at the
second step of the construction, where one characterizes the closure condition d2 d1 =
4
0
0. On the contrary, we are not expecting to see a monomial such as D11 D21
D22
because
it would not make sense to apply d1 after d4 and d0 . Therefore, every monomial
appearing in our calculations, as obtained by left multiplication of two following
differentials, will have the form
· · · Din−t
∂in (∂jn )∂in−1 (∂jn−1 ) · · · ∂in−t (∂jn−t )
m = Dinn (jn ) Din−1
n−t (jn−t )
n−1 (jn−1 )
(6.8)
where the derivatives or indices in parenthesis are optional, depending on the linear
or quadratic nature of the corresponding basic megaform. The idea is to represent
such monomials with a ”semi-commutative” structure. The left side, containing only
139
basic megaforms, can be viewed as commutative. It suffices to give suitable names to
the variables Dinn (jn ) and to pick a term ordering such that they automatically appear
in the desired order. Even without the choice of a suitable term ordering, there would
be no question of ambiguity as long as the superscripts are shown.
Example 6.2. Let us represent the basic megaforms with two indices on CoCoA with
the letter a and with the letter b the basic megaforms with just one index, and
suppose that we give 3 indices to a so that the first represents the degree and the
last two are the effective indices, while for the b’s we just use two indices, one for
3
1
the degree and the second for the index. For instance, D14
D22 D33
will be written
as a[3,1,4]b[2,2]a[1,3,3]. Even in the form b[2,2]a[3,1,4]a[1,3,3] or with
n
any other permutation of the variables, the original basic megaform Dij
could be
reconstructed, considering that the first index of each variable corresponds to n,
while the remaining indices are the subscripts i, j.
Nothing can be done, instead, for the right part of m, since the Dirac operators
could appear in any order and without any a priori structure. We will then represent
such a non commutative object with the list of integers corresponding to the indices
j, from left to right. This way, ∂1 ∂2 becomes the list [1,2] and ∂2 ∂3 ∂12 becomes
[2,3,1,1]. Let us give the following definition:
Definition 6.2. Let k be the number of Dirac operators and let N be an integer
large enough so that the length of the Dirac complex associated is less than N . Let
n
M = Z3 [Di,j
, Dkn | i, j, l = 1 . . . k, n = 0 . . . N ]
(6.9)
be the ring of polynomials whose variables are the basic megaforms of degree at most
140
¯ where D̄ is a squarefree monomial of M,
N . A megamonomial is a pair m = (D̄, ∂)
referred to as the D̄-part of m, and ∂¯ is a (finite) list of integers between 1 and k or,
equivalently, a non commutative monomial in the abstract derivatives, referred to as
¯
the ∂-part
of the megamonomial.
Remark 6.4. The reason why we are limiting the ring of coefficients for the megaforms
to Z3 is that the only elements used in our computations are the differentials dn and
the radial relations [{∂i , ∂j }, ∂` ] = 0. The only mathematical operations on these objects are multiplication of differentials and reductions using either the radial relations
or the relations on megaforms, so at each step every monomial can have a coefficient
only in {−1, 0, 1}.
With this definition of megamonomial, and the consequent representation on
CoCoA as a list of two elements (a monomial in M and a list of integers), it is possible
to perform all the operations required. A differential will be simply a formal sum of
megamonomials. On CoCoA this can be represented by a list of megamonomials, each
carrying its own coefficient, i.e. the sign of D̄. Let us call such a list of megamonomials a megapolynomial. Two megamonomials can be multiplied together using the
formula
(D̄a , ∂¯a ) · (D̄b , ∂¯b ) := (D̄a · D̄b , ∂¯a ∂¯b )
where the product D̄a · D̄b is the usual product in M and ∂¯a ∂¯b stands for the juxtaposition of lists. Thus, two megapolynomials can be multiplied with the usual Cauchy
formula.
Reductions. Particular attention has to be paid when we need to reduce a megapolynomial using either the radial relations among the ∂j ’s or the megaform relations to
141
reduce the D̄’s. In the latter case, such relations on the space of megaforms can
be viewed as polynomials of M and then to reduce a megapolynomial it suffices to
reduce each D̄-part using such polynomials as rewrite rules. Let us clarify this with
a definition.
Definition 6.3. Let f =
P
¯
a (D̄a , ∂a )
be a megapolynomial and let I = (f1 , . . . , fr )
be an ideal of M whose generators have squarefree supports, i.e. each monomial in
fi is squarefree, i = 1 . . . r. Then f can be reduced to f 0 modulo I using the formula
f0 =
X
(NFI (D̄a ), ∂¯a ),
a
where the symbol NFI denotes the normal form of a polynomial with respect to the
ideal I (see [54]) and a given term ordering on M. Furthermore, we say that f 0 is
the mega-normal form of f .
This type of reduction is utilized in a syzygy step, when a megapolynomial has to
be rewritten using the relations on Mk obtained by the closure conditions to get the
syzygies at that step. On the other hand, when considering a closure step one has
¯
to perform a reduction on the ∂-part
of a megamonomial (or megapolynomial) using
the radial relations. We can imagine to do this by rewriting every triple . . . ∂i ∂j ∂` . . .
appearing in ∂¯ using the relation
∂i ∂j ∂` = ∂` ∂j ∂i + ∂` ∂i ∂j − ∂j ∂i ∂` .
(6.10)
Such a relation, however, cannot be applied to rewrite every alignment of indices
(i, j, `) because it would obviously produce an infinite loop. We can then decide to use
it to rewrite . . . ∂i ∂j ∂` . . . only if i is strictly smaller than j and `, with i 6= j 6= ` 6= i.
142
If two of the indices coincide, the radial relation (6.10) becomes
∂i2 ∂j = ∂j ∂i2 .
(6.11)
In this case we can decide to always ”bring squares to the right” no matter the indices
i, j.
Example 6.3. Let us rewrite the monomial ∂1 ∂3 ∂2 ∂3 using the radial relations with
k = 3 as follows:
∂1 ∂3 ∂2 ∂3 = ∂2 ∂1 ∂3 ∂3 + ∂2 ∂3 ∂1 ∂3 − ∂3 ∂1 ∂2 ∂3
and then, since the last new monomial ∂3 ∂1 ∂2 ∂3 contains the alignment ∂1 ∂2 ∂3 in
which the first index is the minimum, we can proceed further and obtain
∂2 ∂1 ∂3 ∂3 + ∂2 ∂3 ∂1 ∂3 − ∂3 ∂1 ∂2 ∂3 = ∂2 ∂1 ∂32 + ∂2 ∂3 ∂1 ∂3 − ∂32 ∂1 ∂2 − ∂32 ∂2 ∂1 + ∂3 ∂2 ∂1 ∂3
and finally, simplifying like terms we get
∂2 ∂1 ∂32 + ∂2 ∂3 ∂1 ∂3 − ∂32 ∂1 ∂2 − ∂32 ∂2 ∂1 + ∂3 ∂2 ∂1 ∂3 = ∂2 ∂3 ∂1 ∂3 − ∂1 ∂2 ∂32 + ∂3 ∂2 ∂1 ∂3 .
The method described in the above example can be more precisely defined.
2
3
Definition 6.4. Let us consider the set of non commutative polynomials rij
and rij`
defined as
2
:= [∂i2 ∂j ] − ∂j ∂i2 ,
rij
i, j = 1 . . . m, i 6= j
3
rij`
:= [∂i ∂j ∂` ] + ∂j ∂i ∂` − ∂` ∂j ∂i − ∂` ∂i ∂j , i, j, ` = 1 . . . m, i = min(i, j, `), i 6= j 6= ` 6= i
where we consider each term in square brackets to be the leading term. Such polynomials will be called radial polynomials. Given a radial polynomial r with leading
143
term t, we define the one step reduction that to each non commutative polynomial
f ∈ Z3 h∂1 , . . . , ∂k i associates the new polynomial
fr := (f /t) · (t − r)
where the division f /t, if possible, is made by dividing f to the right by t. The
iterative process that performs a series of one step reductions starting with f until
each monomials reaches the form
∂i1 · · · ∂iq ∂j21 · · · ∂j2p
(6.12)
where j1 ≤ · · · ≤ jp and il 6= min(il , il+1 , il+2 ) for each l = 1 . . . q − 2, is called
radial normal form of f . The polynomial obtained by this iteration is denoted by
P
NFR (f ). The radial normal form of a megapolynomial m = a (D̄a , ∂¯a ) is defined by
the formula
NFR (m) :=
X
(D̄a , NFR (∂¯a )).
a
Note that if NFR (∂¯a ) is a polynomial, the above formula must be correctly interpreted
as a megapolynomial.
Definition 6.4 allows to perform a closure step. Such a step consists in rewriting
the megapolynomial dn · dn−1 to NFR (dn · dn−1 ), and after this one can group all the
¯
terms that have the same ∂-part.
The coefficients of such terms are then polynomials
in Mk that can be used for the following syzygy step. In the next section, I will
illustrate the algorithm that allows to calculate explicitly the Dirac complex using
the notions of mega-normal form and radial normal form described so far.
144
6.4
The algorithm
Before we present the algorithm that allows the computation of Dirac syzygies using
megaforms, let us make the following remark. At each step of the construction, the
closure condition dn+1 ·dn = 0 has to be characterized by applying the differential dn to
an element gn ∈ Fn . Such element is an unknown function that is recursively defined
by the nonhomogeneous system of equations dn gn = gn+1 . Therefore, besides having
to represent the differentials as megapolynomials, we also have to find a representation
of gn that is suitable for the implementation on CoCoA. Our choice has been to just
put the function gn at the end of each megamonomial, as the last element of the
¯
∂-part.
This requires the introduction of new CoCoA variables g[1]. . . g[N].
2
Example 6.4. Let us represent the megamonomial D13
D21 ∂1 ∂3 ∂2 g1 on CoCoA with
the list [a[2,1,3]b[1,2],[1,3,2,g[1]]]. This has the advantage that we can still
use such a representation to multiply two megamonomials or to find the mega-normal
¯
form. To find the radial normal form, we just need to split the ∂-part
into two parts
([1,2,3] and g[1]) and then perform the radial reduction obviously only on the first
one, appending back g[1] to the list(s) only at the end of the process.
Algorithm 6.1. Let f be a function of F0 . Consider the following iteration:
Input: The megapolynomials dn , n ≥ 0.
Output: From each closure step ci ), an ideal Ii representing the relations satisfied
by the elements of M of degree i. From each syzygy step si ), a set Si containing the
generators of the i-th syzygies of the Dirac complex.
c1 ) Find the radial normal form of the polynomial d1 d0 f , group similar
¯
∂-parts
and collect all the nonzero coefficients as generators of I1 .
145
s1 ) Define g1 := d0 f and find the mega-normal form of d1 g1 with respect
to I1 . Group the similar D̄-parts and collect the nonzero coefficients in
the set S1 .
[. . . ]
cn ) Using the mega-normal form NFIn−1 (dn−1 gn−1 ) from the previous step,
construct the product dn dn−1 gn−1 and find its radial normal form. Collect
¯
the similar ∂-parts
and generate the new ideal In with their nonzero coefficients.
sn ) Define gn := NFIn−1 (dn−1 gn−1 ) and find the mega-normal form of
dn gn with respect to In . Group the similar D̄-parts and collect the nonzero
coefficients in the set Sn .
Iterate the above steps until the set Sn is empty.
The above algorithm has been implemented in CoCoA and run for the case of two,
three, and partly for the case of four Dirac operators. Due to the particular complexity
of the representation chosen, it is difficult to read the outputs and reinterpret them as
megapolynomials or syzygies. Because of the greater significance of the syzygies, in
the following paragraphs I will only present the computations of the sets Si , without
explicitly listing the generators of the ideals Ii . The latter are usually too big and do
not really provide any information on the complex. They just constitute some sort
of ”dual” of the syzygies. To improve the readability of the outcomes, some ad hoc
print functions have been included so that the generators of Si appear in a ”nice”
form. Consider for example the syzygy
∂2 ∂1 g1 − ∂1 ∂2 g2
146
from the first step of the complex for 2 operators. The way CoCoA represents such
relations internally is by using the list [[2,1,g[1]],[1,2,-g[2]]]. However, the
output is presented in the following form: d2d1g[1]-d1d2g[2].
6.4.1
Experiments with 2 Dirac operators
In this paragraph and in the following ones I present the results obtained with the
implementation of Algorithm 6.1 on CoCoA. To simplify the notation, I will not use
superscripts for the basic megaforms as discussed in the previous sections. Let us
begin with the analysis of the Dirac system in two variables
½
∂1 f = g1
.
∂2 f = g2
Defining d0 = D1 ∂1 + D2 ∂2 and d1 = D11 ∂12 + D12 ∂1 ∂2 D21 ∂2 ∂1 + D22 ∂22 we can characterize the condition d1 d0 f = 0 with the relations obtained by annihilating the radial
normal form of the megapolynomial d1 d0 . They are precisely (see [62]):
Dij Di = 0, i, j = 1, 2,
Dii Dj + Djj Di = 0, i, j = 1, 2 i 6= j.
Such relations are exactly the ones returned by CoCoA at the first closure step. Following the notation discussed in the previous section for the names of the variables
we get the following generators for the ideal I1 :
a[1, 1, 1]b[1, 1]
a[1, 2, 1]b[1, 1] + a[1, 1, 1]b[1, 2]
a[1, 1, 2]b[1, 1]
a[1, 2, 2]b[1, 1] + a[1, 1, 2]b[1, 2]
a[1, 2, 1]b[1, 2]
a[1, 2, 2]b[1, 2]
147
Now, defining g := d0 f , we can reduce the megapolynomial d1 g as follows:
D11 D1 ∂12 g1 + D11 D2 ∂12 g2 + D12 D1 ∂1 ∂2 g1 + D12 D1 ∂1 ∂2 g2 +
+D21 D1 ∂2 ∂1 g1 + D21 D1 ∂2 ∂1 g2 + D22 D1 ∂22 g1 + D22 D2 ∂22 g2 =
= D11 D2 (∂12 g2 − ∂2 ∂1 g1 ) + D12 D2 (∂22 g1 − ∂1 ∂2 g2 ) = 0
and from the annihilation of the coefficient of D11 D2 and D12 D2 we get the two
syzygies
½
h1 = ∂12 g2 − ∂2 ∂1 g1
h2 = ∂22 g1 − ∂1 ∂2 g2
which coincide with the ones obtained with CoCoA
h[1] = d1d1g[2] − d2d1g[1]
h[2] = d1d2g[2] − d2d2g[1].
The next closure step provides the last nonzero map of the complex, according to the
relation ∂2 h1 + ∂1 h2 = 0. As a result of the radial reduction of d2 d1 g using CoCoA we
obtain the following set of generators for I2
−a[1, 1, 1]b[1, 2]b[2, 2]
−a[1, 1, 1]b[1, 2]b[2, 1]
−a[1, 1, 2]b[1, 2]b[2, 1] − a[1, 1, 2]b[1, 2]b[2, 2]
−a[1, 1, 2]a[2, 2, 2]b[1, 2]
−a[1, 1, 2]a[2, 2, 1]b[1, 2] − a[1, 1, 1]a[2, 2, 2]b[1, 2]
−a[1, 1, 1]a[2, 2, 1]b[1, 2]
−a[1, 1, 2]a[2, 1, 2]b[1, 2]
−a[1, 1, 2]a[2, 1, 1]b[1, 2] − a[1, 1, 1]a[2, 1, 2]b[1, 2]
−a[1, 1, 1]a[2, 1, 1]b[1, 2]
Because of the choice of the leading terms that CoCoA operates based on the default
term ordering DegRevLex, the previous relations may differ form the one presented in
148
[62], Proposition 4.4, but their number is the same and they still lead to the syzygies
S2 , provided that we make the identification h[2] = −h2 . In this case CoCoA returns
a set of three elements for S2 :
k[1] = −d1d1h[2] + d1d2h[1]
k[2] = −d2d1h[2] + d2d2h[1]
k[3] = −d1h[2] + d2h[1]
of which only the last one is actually sufficient to generate the last syzygy module,
as the first two are simply multiples of the third. Technically, we can conclude at
this point that the following map of the complex is zero because we just have one
generator for the space S2 This can also be empirically checked with CoCoA. After we
eliminate the two dependent syzygies, the set S3 computed at the next syzygy step
is empty. In conclusion, the complex that we have obtained this way in the case of
two Dirac operators is
d0
d1
d2
0 −→ M2 −→ M22 −→ M23 −→ M2 −→ 0.
6.4.2
Experiments with 3 Dirac operators
Let us now consider the inhomogeneous system

 ∂1 f = g1
∂2 f = g2 .

∂3 f = g3
149
and let us utilize algorithm 6.1 to derive the syzygies of the associated complex. At
the first syzygy step, we obtain the 8 relations
h[1] = d1d1g[2] − d2d1g[1]
h[2] = d1d1g[3] − d3d1g[1]
h[3] = d1d2g[2] − d2d2g[1]
h[4] = d1d2g[3] + d1d3g[2] − d2d3g[1] − d3d2g[1]
h[5] = −d1d3g[2] + d2d1g[3] + d2d3g[1] − d3d1g[2]
h[6] = d1d3g[3] − d3d3g[1]
h[7] = d2d2g[3] − d3d2g[2]
h[8] = d2d3g[3] − d3d3g[2]
by comparing these relations to the compatibility conditions (2.30) presented in section 2.4, we see that the two descriptions are equivalent, given the following identifications:
h[1] = −h12
h[5] = −a2
h[2] = −h13
h[6] = h31
h[3] = −h21
h[7] = −h23
h[4] = −a1
h[8] = h32 .
At the second step, the set of relations given by CoCoA consists of 28 dependent
relations
∗(1)
∗(2)
(3)
∗(4)
(5)
(6)
(7)
(8)
(9)
∗(10)
(11)
(12)
(13)
(14)
− d1d1h[3] + d1d2h[1]
− d1d1h[4] + d1d2h[2] + d1d3h[1]
− d1d1h[5] − d1d3h[1] + d2d1h[2] − d3d1h[1]
− d1d1h[6] + d1d3h[2]
− d1d1h[7] + d2d1h[4] − d2d3h[1] − d3d1h[3]
− d1d1h[7] + d2d2h[2] − d3d1h[3]
− d1d1h[8] + d2d1h[6] − d3d3h[1]
− d1d1h[8] + d2d3h[2] − d3d3h[1]
− d1d2h[7] + d2d2h[4] − d2d3h[3] − d3d2h[3]
− d1d2h[8] + d1d3h[7]
− d1d2h[8] + d2d2h[6] − d3d3h[3]
− d1d2h[8] − d2d1h[8] + d2d3h[4] + d2d3h[5] − d3d3h[3]
d1d3h[8] − d2d3h[6] − d3d2h[6] + d3d3h[4]
− d1d3h[8] + d2d3h[6] − d3d1h[8] + d3d3h[5]
150
∗(15)
∗(16)
∗(17)
∗(18)
∗(19)
∗(20)
∗(21)
∗(22)
(23)
(24)
(25)
(26)
(27)
(28)
− d2d1h[3] + d2d2h[1]
− d2d1h[7] + d2d2h[5] + d2d3h[3]
− d2d2h[8] + d2d3h[7]
− d3d1h[3] + d3d2h[1]
− d3d1h[4] + d3d2h[2] + d3d3h[1]
− d3d1h[6] + d3d3h[2]
− d3d1h[7] + d3d2h[5] + d3d3h[3]
− d3d2h[8] + d3d3h[7]
− d1h[3] + d2h[1]
− d1h[4] + d2h[2] + d3h[1]
− d1h[6] + d3h[2]
− d1h[7] + d2h[5] + d3h[3]
− d1h[8] − d2h[6] + d3h[4] + d3h[5]
− d2h[8] + d3h[7]
While some of the quadratic relations are simply multiples of other ones (they have
been indicated with a ∗), there are 16 relations that are not obtained from other ones
with a multiplication by a ∂i . Among these, only 12 constitute the set of independent
syzygies. Unfortunately, while the elimination of the ∗-relations can be done automatically with CoCoA, still the dependence of 4 of the remaining 16 equations have to
be checked manually. For example, we notice that (5) − (6) = d2 ∗ (24) and similar
relations hold for other triples. The final set of 6 quadratic and 6 linear syzygies is
then given by
k[1] = −d1h[3] + d2h[1]
k[2] = −d1h[4] + d2h[2] + d3h[1]
k[3] = −d1h[6] + d3h[2]
k[4] = −d1h[7] + d2h[5] + d3h[3]
k[5] = −d1h[8] − d2h[6] + d3h[4] + d3h[5]
k[6] = −d2h[8] + d3h[7]
k[7] = −d1d1h[5] − d1d3h[1] + d2d1h[2] − d3d1h[1]
k[8] = −d1d1h[7] + d2d2h[2] − d3d1h[3]
k[9] = −d1d1h[8] + d2d3h[2] − d3d3h[1]
k[10] = −d1d2h[7] + d2d2h[4] − d2d3h[3] − d3d2h[3]
k[11] = −d1d2h[8] + d2d2h[6] − d3d3h[3]
k[12] = d1d3h[8] − d2d3h[6] − d3d2h[6] + d3d3h[4]
151
with the following identities that link the above relations to (2.31):
k[1] = −R3
k[7] = T23
k[2] = −S1
k[8] = −U21
k[3] = −R2
k[9] = U31
k[4] = S2
k[10] = −T13
k[5] = −S3
k[11] = −U32
k[6] = −R1
k[12] = −T12 .
As we can see, while the linear syzygies returned by CoCoA are exactly the one described in theorem 2.9, only a few of the quadratic ones appear in the previous set.
In fact, the Uij and Tij are redundant generators, given that can select only Uij and
Tij for i < j. The application of algorithm 6.1 then does not fail in reconstruction at
least a set of 12 independent syzygies. However, one would expect to get all the 18
relations described by the general theory of three Dirac operators. Indeed, note that
some of them are not even present in the output produced by CoCoA before the elimination of the multiples. The reason for this is still unclear. One may argue that the
choice of the radial polynomials and their leading terms influences the computation.
I refer the reader to the next section for a discussion of such problems.
If we now keep the relations k[1] . . . k[12] as elements for S2 , we can find the syzygies
at the third step:
l[1] = d1d1k[4] + d1d3k[1] + d3d1k[1] − d1k[8] + d2k[7]
l[2] = d1d1k[5] − d2d1k[3] + d3d1k[2] − d1k[9] + d3k[7]
l[3] = d1d1k[6] − d2d2k[3] − d1k[11] + d3k[8]
l[4] = d1d2k[6] + d2d1k[6] − d2d2k[5] + d3d2k[4] − d2k[11] + d3k[10]
l[5] = d1d3k[6] − d2d3k[5] + d3d3k[4] + d2k[12] + d3k[11]
l[6] = −d2d2k[2] + d3d2k[1] − d1k[10] + d2k[8]
l[7] = −d2d2k[3] + d3d3k[1] − d1k[11] + d2k[9]
l[8] = −d2d3k[3] − d3d2k[3] + d3d3k[2] + d1k[12] + d3k[9]
which coincides exactly with the set of independent syzygies forecast at this point of
the complex according to the theory. The correspondence table between the ones in
152
theorem 2.9 and the set produced by the machine in this case is hard to recognize.
because the above syzygies make use only of the 12 independent relations of the
previous step, while in the theorem the are described using 18 symbols. Nonetheless,
taking into account the constraints in the quadratic syzygies, we can deduce the
following identities:
l[1] = B12
l[5] = B32
l[2] = −B13
l[6] = −B21
l[3] = C1
l[7] = −C2
l[4] = B23
l[8] = −B31 .
As in the case of the first syzygies, there is one dependent relation, C3 = −C2 − C1
which does not appear in the above set. The next expected compatibility conditions for the complex would be three linear relations given by 2.33. Unfortunately,
CoCoA returns just the two relations
m[1] = d1d1l[5] − d1d3l[3] + d1d3l[7] − d2d1l[8] + d2d3l[2] + d3d1l[7] − d3d3l[1]
m[2] = d1d2l[5] − d1d3l[4] − d2d2l[8] + d2d3l[3] + d3d2l[7] − d3d3l[6]
which in (2.33) correspond to −E2 and E1 respectively. The syzygy E3 is missing for
reasons that, at this point, we cannot really investigate in details. Our best guess
is that, since at the second step the number of relations have been reduced from 28
to 12, some of the relations in the ideal of megaforms I3 have been lost and this
influenced the following syzygy steps. On the other hand, Algorithm 6.1 does not
contain any minimalization step once a set Si is calculated, so the elimination of the
redundant equations is arbitrary and has been done following the results of section
2.4. If one wants to avoid reducing the number of syzygies at the second step, keeping
all the 28 relations given by CoCoA, the third step becomes already impossible to be
153
completed by the machine in a ”reasonable” time. A possible way out, which is a
compromise between keeping all the 28 relations and cutting them down to 12, is to
take into account only the 16 syzygies that are not multiples of other ones:
k[1] = −d1d1h[5] − d1d3h[1] + d2d1h[2] − d3d1h[1]
k[2] = −d1d1h[7] + d2d1h[4] − d2d3h[1] − d3d1h[3]
k[3] = −d1d1h[7] + d2d2h[2] − d3d1h[3]
k[4] = −d1d1h[8] + d2d1h[6] − d3d3h[1]
k[5] = −d1d1h[8] + d2d3h[2] − d3d3h[1]
k[6] = −d1d2h[7] + d2d2h[4] − d2d3h[3] − d3d2h[3]
k[7] = −d1d2h[8] + d2d2h[6] − d3d3h[3]
k[8] = −d1d2h[8] − d2d1h[8] + d2d3h[4] + d2d3h[5] − d3d3h[3]
k[9] = d1d3h[8] − d2d3h[6] − d3d2h[6] + d3d3h[4]
k[10] = −d1d3h[8] + d2d3h[6] − d3d1h[8] + d3d3h[5]
k[11] = −d1h[3] + d2h[1]
k[12] = −d1h[4] + d2h[2] + d3h[1]
k[13] = −d1h[6] + d3h[2]
k[14] = −d1h[7] + d2h[5] + d3h[3]
k[15] = −d1h[8] − d2h[6] + d3h[4] + d3h[5]
k[16] = −d2h[8] + d3h[7]
By exploiting the above relations as the input for the next steps of the algorithm, we
obtain the following third syzygies (after eliminating again the multiple ones):
l[1] = d1d1k[14] − d1d2k[12] + d1d3k[11] + d3d1k[11] − d1k[2] + d2k[1]
l[2] = d1d1k[15] − d1d2k[13] − d2d1k[13] + d3d1k[12] − d1k[4] + d3k[1]
l[3] = d1d1k[16] − d2d2k[13] + d3d2k[12] − d1k[7] + d3k[2]
l[4] = d1d1k[16] − d2d2k[13] − d1k[7] + d3k[3]
l[5] = −d2k[12] − k[2] + k[3]
l[6] = −d2k[13] − k[4] + k[5]
l[7] = −d2k[15] − k[7] + k[8]
154
l[8] = d1d2k[16] + d2d1k[16] − d2d2k[15] + d3d2k[14] − d2k[7] + d3k[6]
l[9] = −d3k[15] + k[9] + k[10]
l[10] = d1d3k[16] − d2d3k[15] + d3d3k[14] + d2k[9] + d3k[7]
l[11] = d1d3k[16] − d2d3k[15] − d3d2k[15] + d3d3k[14] + d2k[9] + d3k[8]
l[12] = d2d1k[7] − d2d2k[4] − d3d1k[6] + d3d2k[2]
l[13] = d2d1k[9] + d2d3k[4] + d3d2k[4] − d3d2k[5] − d3d3k[2] + d3d3k[3]
l[14] = −d2d2k[9] − d2d2k[10] − d3d2k[7] + d3d2k[8]
l[15] = −d2d2k[12] + d3d2k[11] − d1k[6] + d2k[3]
l[16] = −d2d2k[13] + d3d3k[11] − d1k[7] + d2k[5]
l[17] = −d2d3k[9] − d2d3k[10] − d3d3k[7] + d3d3k[8]
l[18] = −d2d3k[13] + d3d3k[12] + d1k[9] + d3k[4]
l[19] = −d2d3k[13] − d3d2k[13] + d3d3k[12] + d1k[9] + d3k[5]
l[20] = d3d2k[11] − d1k[6] + d2k[2]
l[21] = d3d3k[11] − d1k[7] + d2k[4]
Note that it would be possible to find out that some of the 21 relations above are
actually syzygies of the form (2.32), while other ones (like l[5], l[6], l[7] and l[9])
just express the fact that the previous set of equations k[1] . . . k[21] is redundant.
If we continue this way, we get 31 relations in which only the following 15 are not
multiples of each other
m[1] = d1d1l[8] − d1d2l[3] + d1d2l[21] − d2d1l[3] + d2d2l[2] + d3d1l[20] − d3d2l[1]
m[2] = d1d1l[10] − d1d3l[3] + d1d3l[21] − d2d1l[18] + d2d3l[2] + d3d1l[21] − d3d3l[1]
m[3] = −d2l[5] + l[15] − l[20]
m[4] = −d2l[6] + l[16] − l[21]
m[5] = −d2l[18] + d3l[3] − d3l[4] + d3l[16] − d3l[21] + l[13]
m[6] = d2l[21] − d3l[20] + l[12]
m[7] = −d3l[5] − l[3] + l[4]
m[8] = −d3l[6] − l[18] + l[19]
m[9] = −d3l[7] − l[10] + l[11]
m[10] = −d2d2l[3] + d2d2l[4] − d3d2l[15] + d3d2l[20]
m[11] = −d2d2l[18] + d2d2l[19] − d3d2l[16] + d3d2l[21]
m[12] = −d2d2l[18] + d2d3l[16] − d2d3l[21] − d3d3l[15] + d3d3l[20] + d2l[13]
m[13] = −d2d3l[3] + d2d3l[4] − d3d3l[15] + d3d3l[20]
m[14] = −d2d3l[18] + d2d3l[19] − d3d3l[16] + d3d3l[21]
m[15] = −d2l[17] + d3l[14]
155
and finally the fifth syzygies that close the complex:
n[1] = −d2m[5] + m[12] − m[13]
n[2] = −d2m[13] + d3m[10]
n[3] = −d2m[14] + d3m[11]
in conclusion, we have obtained a resolution of the Dirac system associated to the
Dirac operator in three variables that is not minimal, but still has the same length
5 = 2k − 1 as the one described in [63]. The dual of such free resolution is then
d0
d1
d2
d3
d4
d5
0 −→ M3 −→ M33 −→ M38 −→ M316 −→ M321 −→ M315 −→ M33 −→ 0.
6.4.3
Experiments with 4 Dirac operators
The computation of the compatibility conditions of the system

∂1 f



∂2 f
∂3 f



∂4 f
= g1
= g2
= g3
= g4
can be carried out with CoCoA only for the first two syzygy modules. The quadratic
syzygies at the first step are
k[1] = d1d1g[2] − d2d1g[1]
k[2] = d1d1g[3] − d3d1g[1]
k[3] = d1d1g[4] − d4d1g[1]
k[4] = d1d2g[2] − d2d2g[1]
k[5] = d1d2g[3] + d1d3g[2] − d2d3g[1] − d3d2g[1]
k[6] = d1d2g[4] + d1d4g[2] − d2d4g[1] − d4d2g[1]
k[7] = −d1d3g[2] + d2d1g[3] + d2d3g[1] − d3d1g[2]
k[8] = d1d3g[3] − d3d3g[1]
k[9] = d1d3g[4] + d1d4g[3] − d3d4g[1] − d4d3g[1]
k[10] = −d1d4g[2] + d2d1g[4] + d2d4g[1] − d4d1g[2]
k[11] = −d1d4g[3] + d3d1g[4] + d3d4g[1] − d4d1g[3]
156
k[12] = d1d4g[4] − d4d4g[1]
k[13] = d2d2g[3] − d3d2g[2]
k[14] = d2d2g[4] − d4d2g[2]
k[15] = d2d3g[3] − d3d3g[2]
k[16] = d2d3g[4] + d2d4g[3] − d3d4g[2] − d4d3g[2]
k[17] = −d2d4g[3] + d3d2g[4] + d3d4g[2] − d4d2g[3]
k[18] = d2d4g[4] − d4d4g[2]
k[19] = d3d3g[4] − d4d3g[3]
k[20] = d3d4g[4] − d4d4g[3]
which correspond to the 12 relations
∂j ∂i gi − ∂i2 gj = 0,
i, j = 1 . . . 4,
i 6= j
and the 8 relations in the set
{∂j , ∂k }gi − ∂i ∂j gk − ∂i ∂k gj = 0,
i, j, k = 1 . . . 4,
i 6= j 6= k 6= i.
I will not include here the 80 relations forming the set S2 for this case. Let me just
point out that while the computation of the first syzygies required about 300 , this
required several hours. We then decided not to proceed further in the computation
of S3 . Even if the final set of independent relations for S3 should not exceed the
capabilities of the machine we used, still the fact that algorithm 6.1 produces a
redundant set of generators for S3 leads to a tremendous amount of computations.
6.5
Conclusions
In this final section I discuss some of the issues encountered in the computation of
the Dirac complex using megaforms. I also analyze some possible reasons why the
algorithm seems to involve a large amount of computations that lead to a non minimal resolution. Furthermore, I present some possible arguments against the choices
157
implied by this method and possible ways to validate, or invalidate, such arguments.
See also Chapter 8 for some further discussion.
Representation. The ”megamonomial” data type chosen for CoCoA seems to be fair
for a non commutative object. The computations carried out involve multiplications
of megapolynomials, which have the standard complexity of the Cauchy product as
in the commutative case, plus the two types of reduction. Reductions that lead to
the mega-normal form are made with Gröbner Bases techniques so we may assume
that CoCoA is the optimal choice. Radial reductions, on the other hand, could be
performed optimally with other ”low-level” programming languages such as C or C++.
The upcoming version of CoCoA 5.0 should overcome this inconvenience if the routines
written for the implementation of Algorithm 6.1 are translated into C++.
Radial reduction. An objection that could be posed against the computation of the
radial normal form is that the choice of the leading terms in the radial polynomials
is completely arbitrary. The only condition that they guarantee is that the reduction
terminates, leading each monomial to its reduced form (6.12). A different type of final
alignment could be reached, for example, choosing to reduce a triple ∂i ∂j ∂` when i
is the maximum index instead of being the minimum. Also, the squares ∂j2 could be
ordered with decreasing index from left to right, and could be ”brought to the left”
instead of to the right hand side of each monomial. All these different choices have
been tested with CoCoA but they have not produced any significant difference in performance. The ideal of megaform relations In may vary, but the syzygies obtained at
each step are exactly the same. This leads to assume that the generation of the sets
158
Sn in the algorithm is actually independent of the choice of the radial polynomials
(6.11) and (6.10).
Megaform reduction. A similar argument holds for the way we calculate the meganormal form of a megapolynomial. The computations shown in [62] do not make use
of any particular structure on the algebra of megaforms. In this chapter, we are giving to such a space a structure of subspace of a polynomial ring, and we decide to
perform the reduction by using Gröbner Bases. In particular, one may think that the
results of our computations depend on the choice of the term ordering with which
we endow the ring M. Using different standard term orderings in CoCoA, we have
indeed obtained different representations for the spaces Fn , since the terms that get
reduced in dn dn−1 depend on the leading terms of the generators of In . However, the
set of relations Sn obtained with the subsequent syzygy step has never been affected
by a different representation of In .
Minimality. As Sections 6.4.2 and 6.4.3 have shown, a crucial role in our method is
played by the choice of generators for Sn . It seems that the choice of a minimal set
of generators, besides having the disadvantage of having to be performed ”by hand”,
is also not the right way to produce all the maps in the complex. Unfortunately,
keeping all the redundant relations that CoCoA returns at each syzygy step is not a
option, because the computation of Sn becomes already too heavy for n = 3. This
issue has still to be investigated properly and it is probably the key to decide whether
or not the method of megaforms is computationally ”useful” for the construction of
Dirac syzygies and, more in general, for the effective computation of noncommutative
159
syzygies.
Complexity. When dealing with the calculation of a free resolution, one also has
to take into account the complexity of the problem. For the case of an ideal in a
commutative ring of polynomials, and more in general for any Gröbner Basis related
problem, the nature of the complexity is doubly exponential as [7] has shown. This
may or may not be the particular complexity for the calculation in a specific case, but
it gives an upper bound. Let us now focus on the computations involved in Algorithm
6.1. The Cauchy product of two differentials has just polynomial complexity in the
length of dn , which is n2 + n. So this step only requires O(n4 ) operations. When we
consider the radial reduction we incur in an exponential complexity. Let us suppose
that we have to reduce a monomial of the type ∂i1 · · · ∂it . Then, possibly, each of the
triples
∂ij ∂ij+1 ∂ij+2 ,
j = 1...t − 2
has to be reduced using the radial polynomials (6.10) and then produces 3 new monomials that we keep reducing. We may then estimate the worst case as having a
complexity of order (3t ). Experimental evidence of this is shown by CoCoA since
the computation of the normal form is quite slow (see [28] for an updated chart of
CPU times). A possible optimization of this particular step could then lead to improvements. Finally, the calculation of the mega-normal form has the usual doubly
exponential complexity of any Gröbner Basis calculation. Moreover, the DegRevLex
Gröbner Basis of In has cardinality approximately four times bigger than the cardinality of In itself as direct computation with CoCoA show. In conclusion, both from the
experiments we have run and from general considerations on complexity for Gröbner
Bases, this step seems to be the heaviest one.
Chapter 7: Noetherian operators
7.1
Fundamental Principle
In 1970 Ehrenpreis [35] and (independently) Palamodov [60] discovered an important
representation theorem for solutions of systems of linear constant coefficients partial
differential equations. This result is known as the Ehrenpreis–Palamodov Fundamental Principle and its simplest case (one differential equation in the space C ∞ (Rn ) of
complex valued differentiable functions on Rn ) may be stated as follows.
Theorem 7.1. Let p(D) be a linear constant coefficient partial differential operator
in n variables. Then there are algebraic varieties V1 , . . . , Vt in Cn and constant coefficients differential operators ∂1 , . . . , ∂t , such that every function f ∈ C ∞ (Rn ) satisfying
p(D)f = 0
can be represented as
f (x) =
t Z
X
j=1
∂j (eix·z )dνj (z),
(7.1)
Vj
for suitable Radon measures dνj (the only condition on such measures is that they
ensure the convergence of the integrals in (7.1)).
The operators ∂1 , . . . , ∂t are called, in Palamodov’s terminology, noetherian operators because their construction relies essentially on a theorem of M. Noether on
160
161
a membership criterion for polynomial submodules (see e.g. [60] pp.161, 162). The
nature of the original proof of the Fundamental Principle is essentially existential
and therefore the question of the explicit construction of such operators is of great
interest whenever we consider a concrete application of the Fundamental Principle.
This is particularly relevant when one considers the far reaching generalization of this
result to the case in which the operator p(D) is replaced by a rectangular matrix of
operators.
The idea to use Gröbner Bases for the problem of the construction of a ”dual
basis” for and ideal goes back to [58]. This chapter presents and expands the results
from our paper [29]. Our aim is to build on some recent results in the construction
of noetherian operators [47, 48, 56, 59, 69] and to provide some new algorithms which
allow the automatic construction of these operators at least in some rather large class
of cases. In the case of zerodimensional ideals we perform an optimization of the
algorithm as suggested after a conversation with B. Sturmfels.
Let us stress the fact that, as always, we are concerned with providing algorithms
which can be implemented on existing computer algebra packages such as CoCoA [21]
and Singular [46]. This means in particular that we will be able to utilize only partially
some of the existing ”explicit” constructions such as the one provided in [59], because
such constructions still present unresolved issues, mostly related to the fact that some
algebraic tools (such as the field of fractions over a polynomial ring and the algebraic
closure of a subring) are not yet implemented on any software package.
In sections 7.2 we review the fundamental algebraic tool of primary decomposition
which will be needed for the construction and discuss a possible implementation of
an algorithm [54] for the zerodimensional case. Section 7.3 is devoted to a description
162
of the Fundamental Principle and its relation with primary decomposition and the
concept of multiplicity variety. The core of this chapter is section 7.4 where we deal
with the case of zerodimensional ideals and where we present two explicit algorithms
to construct the noetherian operators. Section 7.5 indicates how to extend one of
the algorithms to the case of zerodimensional modules, and section 7.6 deals with the
case of ideals of higher dimension. The following section 7.7 uses standard monomials
to optimize the computation of noetherian operators in the zerodimensional case. We
conclude the chapter with the calculation of the complexity of our algorithms in
section 7.8.
Executable versions of the algorithms discussed in this chapter have been explicitly
written for CoCoA and are freely available at our CoAlA web page [28].
7.2
Primary decomposition
In the following section I present an overview of the theory of primary decomposition
of ideals. A reference for the missing proofs is [36]. To keep the setting as general as
possible, we will now work with a commutative ring R with unit 1, not necessarily
a ring of polynomials. An ideal of R will be denoted by I and M will denote an
R-module. If R is the ring of integers Z or the polynomial ring in one indeterminate
K[x], the unique factorization property and the fact that every ideal is principal
provide a decomposition of each ideal into ”simple” pieces. For instance, if (n) is an
ideal in Z, then the prime factorization n = pa11 · · · pas s results in the decomposition
(n) = (p1 )a1 ∩ . . . ∩ (ps )as
163
and similarly the ideal (f (x)) of K[x] decomposes into
(f ) = (p1 (x))a1 ∩ . . . ∩ (ps (x))as
where the pi (x) are the irreducible factors of f . In the case of polynomial rings in
several variables, and in general for rings in which each ideal is finitely generated but
not necessarily principal, it is still possible to define a decomposition of ideals that
mimics the decomposition in a Principal Ideal Domain (PID). Let us recall some basic
definitions of commutative algebra that will allow to introduce the topic of primary
decomposition.
Definition 7.1. An element x ∈ R is a zerodivisor if there exists y 6= 0 in R such
that xy = 0, it is invertible if there is y ∈ R such that xy = 1, and it is nilpotent if
there is an integer n > 1 such that xn = 0. The set of all zerodivisors of R is denoted
by Z(R) and the set of all nilpotent elements with Nil(R).
7.2.1
Primary ideals and associated primes
Definition 7.2. An ideal I of R is said irreducible if given a decomposition I = J ∩K
where J and K are two ideals of R, then I = J or I = K. An ideal P is prime if the
quotient R/P has no zerodivisors different from zero (i.e. it is a domain). A maximal
ideal is an ideal which is maximal with respect to inclusion, or equivalently an ideal
m such that R/m is a field. The set of all prime ideals of R is called the spectrum
of R and it is denoted by Spec(R). The set of all maximal ideals will be denoted by
Max(R), and the set of all irreducible ideals with Irr(R).
164
Definition 7.3. Given an ideal I of R, the radical of I is the ideal
√
I = {x ∈ R | xn ∈ I for some n ∈ N},
and I is said to be radical if I =
√
I.
The following facts are easy consequences of the definitions.
Proposition 7.1. Given an ideal I of R we have
p
1) (0) = Nil(R) ⊆ Z(R),
2)
√
I = Nil(R/I) ⊆ Z(R/I),
3) for every n ∈ N,
4) if P is prime,
√
√
In =
√
I,
P = P.
Note that the converse of 4) in Proposition 7.1 is false in general as the example
P = (xy) ⊆ K[x, y] shows. Fact 3) says that we lose ”multiplicities” once we consider
the radical of an ideal. The most important definition of this section is of course the
following:
Definition 7.4. An ideal Q of R is called primary if for every pair of elements x, y
in R such that xy ∈ Q we have either x ∈ Q or y n ∈ Q for some positive integer n.
Equivalently, Q is primary if Z(R/Q) ⊆ Nil(R/Q). The set of all primary ideals of
R will be denoted by Prim(R). Since the radical of a primary ideal is prime, we say
√
that Q is P-prime, with P a prime ideal of R, if Q = P.
Note that in Z the prime ideals are exactly of the form (pi ) , with pi a prime
integer, and primary ideals are powers of prime ideals (and viceversa these are all the
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primary ideals in Z). Primary ideals for a ring then constitute the equivalent of the
ideals (pi )ai in Z, and prime ideals play the role of the ideals (pi ). As in the case of
Z (or K[x]), a power of a prime P is always P-primary. However, it is not true in
general that every primary ideal is a power of a prime, like the example Q = (x, y 2 )
in K[x, y] shows. This last property is only true if R is a PID. We can summarize
some properties of primary ideals that will be crucial for the primary decomposition
in the following proposition.
Proposition 7.2. Let R be a ring. Then
1) if R is noetherian and Q is P-primary, there exists a positive integer n such that
P n ⊆ Q ⊆ P,
2) if Q1 , . . . , Qs are P-primary,
Ts
i=1
Qi is also P-primary.
For a primary ideal Q, the integer n of property 1) in Proposition 7.2 is often
called the index of P in Q. For a polynomial ideal, this notion coincides with the
notion of multiplicity of the affine algebra R/Q.
The spectrum of a ring. Let us now focus on the set of all prime ideals of a ring
R, the spectrum Spec(R). It contains, so to speak, the geometry of the ring R in the
sense that ”irreducible varieties” correspond to prime ideals in R. This statement has
to be specified with the introduction of the notion of noetherian rings. Noetherian
rings are in a way the most natural rings in which to study geometry since their ideals
are finitely generated and, like in the case of a ring of polynomials, they can represent
algebraic varieties in an opportune affine space.
Definition 7.5. A ring R is said to be noetherian if every infinite ascending chain of
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ideals
I0 = (0) ⊆ I1 ⊆ I2 ⊆ I3 ⊆ . . . ⊆ In ⊆ . . .
is eventually constant, i.e. In = In+1 for every n greater than some N ∈ N. Equivalently, it is possible to show that R is noetherian if and only if every ideal of I is
finitely generated. For an R-module M the definition is similar: M is noetherian if
the ascending chain condition (acc) holds on submodules, i.e. if
M0 ⊆ M1 ⊆ M2 ⊆ M3 ⊆ . . . ⊆ Mn ⊆ . . .
is eventually constant. Equivalently, M is noetherian if R is noetherian and every
submodule of M is finitely generated.
In a ring R the following inclusions hold for the set of ideals defined so far:
Max(R) ⊂ Spec(R) ⊂ Irr(R) ⊂ Prim(R)
(7.2)
where the last inclusion is the only one that requires R to be noetherian. It is easy
to see that the inclusions in (7.2) are strict in general, as the following examples in
K[x, y] show.
Example 7.1. The ideal (x) in K[x, y] is prime but not maximal, (x2 ) is irreducible
but not prime and (x, y)2 = (x2 , y) ∩ (x, y 2 ) is primary (it is a power of the maximal
ideal (x, y)) but not irreducible.
The set Spec(R) can be endowed with a special topology called Zarisky topology,
whose closed sets, parameterized by the ideals of R, are
V(I) = {P ∈ Spec(R) | P ⊇ I},
I ∈ Spec(R).
167
The set V(I) is often referred to as the variety associated to I, since in the case of
a ring of polynomials K[x1 , . . . , xn ] that is exactly what it corresponds to. It is the
collection of all the irreducible components of the algebraic variety
V (I) = {x ∈ Kn | f (x) = 0 for all f ∈ I}.
Let us now consider an R-module M . In particular we are interested in the case
in which I is an ideal of R and M = R/I. The following sets are relevant for the
geometry of the spectrum of R.
Definition 7.6. The associated primes of an R module M are the elements of
Spec(R) of the form P = 0 :R hmi for some m in M . The set of all associated primes
of M is indicated with Ass(M ). In particular, if M = R/I we put
Ass(I) = {P ∈ Spec(R) | P = I :R x , x ∈ R}.
The minimal elements with respect to inclusion in Ass(M ) are called minimal primes
of M and they are indicated with Min(M ). If M = R/I we set
Min(I) = {P ∈ Spec(R) , P ⊇ I | I ( P 0 ⊆ P ⇒ P 0 = P}.
Finally, the support of M , denoted by Supp(M ), is the set of all primes P such that
the localization (see [57]) of M at P is not trivial. In other words, P is in the support
of M if there are regular functions on P, i.e. MP 6= 0. For R/I we define
Supp(I) = {P ∈ Spec(R) , P ⊇ I | (R/I)P 6= 0}.
The following properties of the support of M are easy to verify from the definition.
Proposition 7.3. Let M be an R-module. Then
168
1) if M = hmi and I = 0 : M then M ' R/I and Supp(M ) = V(I)
S
2) if M = ⊕j Mj then Supp(M ) = j Supp(Mj )
3) if M is finitely generated over R then Supp(M ) = V(0 : M )
4) if P ∈Supp(M ) then V(P) ⊆Supp(M ).
The geometry of M when M is noetherian is then encapsulated in Supp(M ), as
it represents a generalization of the Zarisky topology (of closed sets) associated to an
ideal, thanks to 3). It is not difficult to prove at this point that for an R module M
we have
Supp(M ) =
[
[
V(P) =
P∈Ass(M )
V(P),
P∈Min(M )
from which we get the following inclusions:
Min(M ) ⊂ Ass(M ) ⊂ Supp(M ) ⊂ Spec(R).
When M is noetherian, the set of all the zerodivisors Z(M ) is a closed subset of the
support which is given by the union of all the associated primes:
Z(M ) =
[
P.
P∈Ass(M )
It is the noetherianity of M that guarantees that the union is finite and non empty.
7.2.2
Existence of an irreducible decomposition
The associated primes play an important role in describing the geometry of an R
module, as they generate its support. Let us now see how they can be used to
construct a primary decomposition of an ideal I of R. First, we give a general
169
definition of primary decomposition that does not require I to be finitely generated.
Definition 7.7. Let I be and ideal of R, let s be a positive integer, let Q1 , . . . , Qs
be primary ideals and let P1 , . . . , Ps be prime ideals of R such that Qi is Pi -primary
for every i = 1 . . . s. Then
I = Q1 ∩ . . . ∩ Qs
(7.3)
is a primary decomposition of I with associated primes Pi . Such decomposition is
T
called minimal if the prime ideals Pi are distinct and Qi + j6=i Qj for every i =
1 . . . s.
The following famous result is due to Emmy Noether and guarantees the existence
of a primary decomposition for every noetherian ring, although the proof is highly
non-constructive.
Theorem 7.2. Let R be a noetherian ring and let I be an ideal of R. Then there
exists a minimal primary decomposition (7.3).
Sketch of the proof. First one proves the existence of an irreducible decomposition
I = Q1 ∩ . . . ∩ Qs ,
i.e. a decomposition in which the Qi are irreducible ideals. This is done thanks to the
noetherianity of R and the use of Zorn Lemma. Since R is noetherian every irreducible
ideal is then primary as from inclusion (7.2). The minimality follows by just eliminating the redundant primary components from the decomposition using part 2) of
Proposition 7.2 .
¤
170
Besides being non-constructive, Theorem 7.2 does not provide a unique decomposition for the ideal I. In fact, uniqueness cannot be achieved as the following examples
in K[x, y] demonstrate.
Example 7.2. The ideal I = (x2 y) admits the primary decompositions
I = (x2 ) ∩ (y) ∩ (x, y) = (x2 ) ∩ (y)
It is clear that in this case the non minimality is due to a choice of a redundant
primary component, (x, y) whose associated prime - the ideal (x, y) itself - is ”too
large”. However, even if we just admit a minimal number of primary components, we
can find an example in which the decomposition is not unique. The ideal J = (x2 , xy)
can be decomposed as follows
J = (x) ∩ (x, y)2 = (x) ∩ (x2 , y) = (x) ∩ (x2 , y + λx)
for any λ in K. Nonetheless, while the primary components are different, the associated primes are always (x) and (x, y). It is not a case that the arbitrary choice is
possible for the (x, y)-primary component while (x) is fixed in all the above decompositions. The following result will specify what kind of uniqueness we can expect
from a minimal decomposition.
Proposition 7.4. Let I be an ideal of a noetherian ring R and let
I = Q1 ∩ . . . ∩ Qs = Q01 ∩ . . . ∩ Q0r
be two minimal primary decompositions with Pi =
√
Qi and Pi0 =
p
Q0i . Then s = r
171
and the prime ideals are actually the associated primes
Ass(I) = {P1 , . . . , Ps } = {P1 , . . . , Pr }.
Moreover if Min(I) = {P1 , . . . , Pk } is the subset of minimal primes of I then, modulo
a sorting of the indices, we have that Qi = Q0i for all i = 1 . . . k.
Looking back at Example 7.2, the primary (x), which is also a minimal associated
prime, corresponds to the y axis, which is a maximal irreducible component (the
only one) of the algebraic set V (J). Therefore it is always part of any primary
decomposition, according to Proposition 7.4. On the other hand, the choice of the
primary associated to the origin is arbitrary, the associated prime (x, y) not being
minimal.
7.2.3
An algorithm for primary decomposition
It is quite surprising that even thought Emmy Noether’s proof relies on Zorn Lemma,
it is still possible to find algorithms for the computation of a primary decomposition.
Probably one of the most famous is [42]. This and other variations have been implemented for ideals and modules over the ring of polynomials on the latest version
of Singular and Macaulay2. I refer the reader to the consultation of the manual of
the Singular library primdec.lib for further information. A complete treatment of
primary decomposition from the computer algebra point of view is not part of my
intent. I will just present here a simple adaptation to the case of zerodimensional
ideals that can be easily be implemented on CoCoA. We are looking forward to have
an implementation of the algorithms in [42] for the upcoming version of CoCoA. From
now one we will assume that the field K has characteristic zero (the actual use that
172
we do of primary decomposition for differential equations is when K = C as we will
see in the next sections) and that R is a ring of polynomials.
Definition 7.8. Let I be an ideal of R = K[x1 , . . . , xn ] where K is a field and K̄ is
its algebraic closure. Let σ be a term ordering. We say that I is zerodimensional if it
satisfies any of the following equivalent conditions:
a) I is contained in only finitely many maximal ideals of R.
b) I ∩ K[xj ] 6= (0) for all j = 1 . . . n.
c) The geometric variety V (I) is a finite union of points in K̄n .
d) The K-vector space R/I is finite-dimensional.
e) The set Tn \LTσ (I) is finite.
f) The Krull dimension of the K-algebra R/I is zero.
Remark 7.1. It is immediate to show that conditions a) through f ) are equivalent,
see for example [54], prop. 3.7.1. If we put K = C, the finite number dimC (R/I) in
condition d) represents exactly the number of points, counted with their multiplicity,
of the variety associated to I. It is a very important numeric invariant since when I is
primary it corresponds to the index defined in 7.4. It is also the number of elements
of the monomial set of condition e). In fact, such ”residual” monomials span the
C–vector space R/I.
Given a zerodimensional ideal I, we assume that we are able to compute, through
standard Gröbner Basis calculations, the following objects (see again [54], section 3.7
for definitions and procedures):
1) the radical ideal
√
I using squarefree polynomials,
2) the elimination ideal I ∩ K[xi ] with respect to any indeterminate xi ,
173
3) the factorization of any univariate polynomial p ∈ K[xi ],
4) the saturation of I with respect to a maximal ideal m, I :R m∞ .
Another key element in the explicit construction of a zerodimensional primary
decomposition is the so called normal position. We will come back to this concept in
a more general setting in section 7.6. For now, let us give a definition for zerodimensional ideals.
Definition 7.9. Let I be a zerodimensional radical ideal in R = K[x1 , . . . , xn ] and
let xi be an indeterminate. Let us denote by K̄ the algebraic closure of K. We say
that I is in xi -normal position if any two zeroes of I in K̄n are distinct.
It is always possible, with a change of coordinates of the type
x0i = xi ,
i < n,
x0n = xn − c1 x1 − . . . − cn−1 xn−1
to put a radical zerodimensional ideal into xn -normal position. Note that the choice
of the scalars ci ∈ K is generic. With the package available at [28] one can use the
command Coala.NormalPos(I) to put an ideal I in normal position. Given all these
ingredients, we can now present an effective algorithm for the computation of the
primary decomposition of a zerodimensional ideal.
Algorithm 7.1. Let I be a zerodimensional ideal of R = K[x1 , . . . , xn ]. Consider the
following list of instructions:
1) Compute the radical ideal of I, J :=
√
I
2) Put J in xn -normal position. Rename it J.
3) Compute, via elimination, the ideal Jn := J ∩ K[xn ] and let p(xn ) be its monic
174
generator.
4) Factor p into irreducible factors p(xn ) = f1 (xn ) · · · fs (xn ).
5) Construct the maximal ideals mi := J + (fi ) for every i = 1 . . . s.
6) Define Qi := J : (J : m∞
i ).
The above procedure calculates a primary decomposition I = Q1 ∩ . . . ∩ Qs of the ideal
I whose associated (maximal) primes are Ass(I) = {m1 , . . . , ms }.
n
Proof. The radical ideal J is an ideal of points in K of multiplicity one. Since it is in
xn -normal position, all the roots of the monic generator p(xn ) are distinct and simple
n
in K . Each factor fi of p then has K-conjugate roots. In particular this implies that
J + (fi ) is maximal for every i = 1 . . . n since R/J + (fi ) is isomorphic to K. It is
then a standard procedure called isolation [42] to construct the primary components
as the colon ideals of the saturations in 6).
Example 7.3. In C[x, y] let us consider the ideal I = (f1 , f2 , f3 ) where
f1 = 189y 3 + 150xy + 252y 2 − 100x − 513y + 174,
f2 = 100x2 + 420xy + 441y 2 − 180x − 378y + 81,
f3 = 9xy 2 − 12xy − 27y 2 + 4x + 36y − 12.
Applying the function Coala.PrimDec to the ideal I we obtain the following output
[[Ideal(y2 , x2 + y), Ideal(x, y)],
[Ideal(y2 − 4/3y + 4/9, xy − 2/3x + 1/2y − 1/3, x2 + x + 1/4), Ideal(2x + 1, 3y − 2)]]
which means that the primary decomposition is
I = (y 2 , x2 + y) ∩ (4x2 + 4x + 1, 6xy − 4x + 3y − 2, 9y 2 − 12y + 4)
175
and the associated maximal ideals are
m1 = (x, y),
m2 = (2x + 1, 3y − 2).
It is clear from Example 7.3 that the geometric nature of a zerodimensional ideal
is captured by the collection of the maximal associated primes. It follows indeed
√
from the construction presented in Algorithm 7.1 that the radical ideal I is the
intersection of all such maximal ideals. Since the radical ideal, according to Hilbert’s
Nullstellensatz, is geometrically indistinguishable from the ideal, this says that the
primary components Qi actually contain more ”algebraic information” than the maximal ideals alone. In the case of a polynomial ring, they also describe the multiplicity
of each irreducible component. This concept of multiplicity is the object of the next
section.
7.3
Multiplicity variety
Definition 7.10. Let Vj be (not necessarily distinct) algebraic varieties in Cn and
let ∂j ∈ An be differential operators with polynomial coefficients, j = 1, . . . , t. We
say that the collection
V = {(V1 , ∂1 ); (V2 , ∂2 ); . . . ; (Vt , ∂t )}
is a multiplicity variety.
Theorem 7.3. Let I be an ideal of R. There exists a multiplicity variety V such that
a polynomial f belongs to I if and only if ∂j f|Vj = 0 for every j = 1, . . . , t.
The result we have just quoted is much stronger than the classic Nullstellensatz
and provides a nice differential characterization of the ideal I. In the case of a radical
176
ideal, it is obvious that the multiplicity variety reduces to {(V (I), id)} and this is
basically an equivalent statement of the Nullstellensatz. The goal of our work will be
to construct explicitly the multiplicity variety associated to an ideal. In particular
we want to find a way to compute the operators ∂j .
As Ehrenpreis showed, theorem 7.3 is essentially equivalent to his Fundamental
Principle, which we now state for the case of non–principal ideals.
Theorem 7.4. Let p1 (D), . . . , pr (D) be linear constant coefficients partial differential
operators in n variables. Then there are algebraic varieties V1 , . . . , Vt in Cn and
differential operators ∂1 , . . . , ∂t ∈ An , such that every function f ∈ C ∞ (Rn ) satisfying
p1 (D)f = . . . = pr (D)f = 0
can be represented as
f (x) =
t Z
X
j=1
∂j (eix·z )dνj (z),
(7.4)
Vj
for suitable Radon measures dνj .
Remark 7.2. A major motivation for our work is the fact that the Fundamental
Principle can be extended to the case in which we are considering matrices of differential operators. Specifically, one can prove a version of Theorem 7.4 for the case in
which we consider an r1 × r0 matrix P = [Pij (D)] of linear constant coefficients differential operators and we are interested in characterizing the space of differentiable
177
solutions (f1 , . . . , fr0 ) to the system
r1
X
Pij (D)fj = 0.
i=1
The result which one obtains (see [35, 60]) formally looks like Theorem 7.4 but the
construction of the noetherian operators is considerably more involved. Our particular interest originated with our study of the Cauchy–Fueter system and its many
variations (see [25] and references therein).
Let us begin by discussing how to find the algebraic sets V1 , . . . , Vt in Theorem
7.4. from the previous section we know that every ideal in a noetherian ring can be
viewed as an intersection of ”simpler parts” which are primary ideals:
I = Q1 ∩ · · · ∩ Qt .
In the case of a polynomial ring R, each Qj represents a geometric component of
the variety V (I) and carries its multiplicity. While the decomposing in Z and k [x]
can be constructed ”by hand”, at least in theory, the proof of the existence of a
primary decomposition is non constructive, as it makes use of Zorn Lemma and the
noetherianity of R. Since the early development of computer algebra, there have been
some successful attempts to make such construction explicit, in order to implement
a primary decomposition algorithm on a computer. Through the computation of a
primary decomposition, then, it is easy to come up with a set of algebraic varieties
V1 , . . . , Vt by simply putting Vj := V (Qj ). This describes completely V (I) as a
set, since V (I) = V1 ∪ · · · ∪ Vt , leaving the information on the multiplicity of each
component to the operators ∂j ’s.
178
Remark 7.3. As simple as it seems, the computation of a primary decomposition
is not just something that we can leave to a machine without some significant intervention. Consider, for example, the principal ideal I = (x2 + y 2 ) in C[x, y]. It is
immediate to show that in this case the decomposition is
I = (x + iy) ∩ (x − iy),
the variety V (I) being the union of two complex lines (note that in this case we can
talk about the primary decomposition since some uniqueness arguments hold in the
case of components of maximal dimension). However, working with C as ground field
can lead to problems when trying to perform the decomposition with the use of a
computer. Suppose that we want to use Singular and define the ideal I on Singular
by entering the command
ideal i=x2+y2;
primdecGTZ(i);
(note that the library primdec.lib has first to be loaded on Singular in order to use
the function primdecGTZ). The only result that we get is I itself. In fact Singular
assumes that we are using the ring of polynomials Q[x, y] as the base ring for I, and
in this case such an ideal happens to be primary. This fact shows evidence of the
limitations of computer algebra: a polynomial ring with a non-computable coefficient
field K (such as K = C) cannot be defined on any computer algebra software package.
Hence when calculating a primary decomposition and more in general anything that
is related to factorization and system solving over K[x1 , . . . , xn ] we have too keep in
mind this constraint. The only rings that we may use are finite algebraic extensions
of the ground field Q, which are still computable. The problem is that one has
179
to know in advance which elements to add to the field! In the previous example,
for instance, the (minimal) environment in which a correct decomposition can be
achieved is Q(i)[x, y], i.e. we need to extend the rational numbers with the root of −1.
This task can be more difficult than the computation of the primary decomposition
itself, especially with more complicated cases such as modules or ideals generated by
more than one element. Ultimately, the main problems associated to computing a
primary decomposition is to handle algebraic field extensions and to solve polynomial
equations or system of equations. We refer the reader to [56], Section 2, where some
discussion on this topic is presented.
Remark 7.4. An alternative and natural approach to the problem of performing
computations in C[x, y] could be to consider machine-approximation of real numbers
via floating point numbers, hence working in a field like Q + iQ and allowing approximation errors. Both CoCoA and Singular allow this feature with some particular
setting commands. However this goes beyond our scope and we will not follow this
approach.
Now that we briefly discussed the computations of the algebraic sets Vj ’s, we are
ready to show the calculation of the noetherian operators in some explicit examples.
Let us begin with the essentially trivial case of an ideal I = (p(x)) ⊆ C[x]. Its primary
decomposition follows easily from the factorization of p into prime factors:
I = (q1 (x)α1 ) ∩ · · · ∩ (qt (x)αt ),
αj ∈ N.
The problem is now reduced to the computation of the noetherian operators attached
to each primary component of the form Q = (q(x)α ) with α ∈ N. Since C is algebraically closed, q(x) = x − a for some a in C. Then the differential conditions that
180
characterize the membership of a polynomial h(x) to Q are trivially:
h ∈ Q ⇐⇒
∂β
h(a) = 0 ,
∂x β
β = 1 ...α − 1
The result can be summarized in a proposition
Proposition 7.5. Let I = (p(x)) be an ideal of C[x], and let p = q1α1 · · · qtαt be the
factorization of its generator. Then
V = {(q1 (x) = 0, id), . . . , (q1 (x) = 0, ∂xα1 −1 ), . . . , (qt (x) = 0, id), . . . , (qt (x) = 0, ∂xαt −1 )}
is a multiplicity variety for I.
Remark 7.5. In this case the Fundamental Principle is nothing but Euler’s well
known exponential representation for solutions of linear constant coefficients ordinary
differential equations.
As shown in [60], the same construction as in Proposition 7.5 can be performed
in the multivariate case n > 1, if the ideal I is principal, i.e. I = (p(x1 , . . . , xn )).
Any prime factor q of p gives rise to the primary component (q α ) of I. The operators
attached to this component are again powers of the derivative with respect to the
variable xj , provided that with a change of coordinates we make sure that its maximal
deg(q)
power xj
is the leading term of q with respect to any degree compatible term
ordering (see [60], prop. 3 page 131). This condition, which guarantees that the ideal
is in normal position with respect to xj , was not needed in the univariate case since
it holds trivially.
Example 7.4. Consider the principal ideal I = (x2 y 2 z 2 − 2xyz 2 + z 2 ) ⊂ C[x, y, z].
The primary decomposition, that can be easily performed with Singular, gives I =
181
(z)2 ∩ (xy − 1)2 , i.e. I represents the union of the xy-plane, with multiplicity 2, with
the hyperbolic cylinder xy = 1 again counted twice. While the first component has
the operators id and ∂z associated to it (Proposition 7.5), the cylinder has first to
be rotated in order to be put in normal position with respect to one variable, say x.
If one performs the change of coordinates X = 12 (x + y); Y = 21 (x − y), one gets the
new polynomial (X 2 − Y 2 − 1)2 which is in normal position with respect to X and so
the noetherian operators attached to the cylinder are id and ∂X. Returning to the
original variables x, y we see that a multiplicity variety for I is
1
V = {(z = 0, id), (z = 0, ∂z), (xy = 1, id), (xy = 1, (∂x + ∂y))}
2
7.4
Noetherian operators for a zerodimensional ideal
The examples of the previous sections only required the computation of a primary
decomposition and, at most, an easy change of coordinates. Such steps are usually
necessary even in more complicated cases. Thus, new computational tools have to be
introduced in order to address the case of a zerodimensional ideal in C[x1 , . . . , xn ].
It corresponds to an ideal of points in Cn since C is algebraically closed. From
now on we assume that I is a primary zerodimensional ideal. Since a zerodimensional
primary ideal is associated to a single point of the variety V (I) we can always assume,
with a change of coordinates, that V (I) = {(0, . . . , 0)}, or equivalently that
(x1 , . . . , xn ).
√
I =
182
7.4.1
Closed differential conditions
A first complete description of the differential condition characterizing a zerodimensional primary ideal centered in zero has been done in [56]. The importance of this
work is not only that an explicit procedure to find the noetherian operators has been
given for this case, but rather that some important theoretical aspects have been
pointed out. Since we will use some of the results from [56], let us briefly recall the
main notations and definitions of that paper. We will denote by D(i1 , . . . , in ) : R → R
the differential operator:
D(i1 , . . . , in ) =
1
∂xi1 · · · ∂xinn ,
i1 ! · · · in ! 1
ij ∈ N , for all j = 1, . . . , n,
or, alternatively, if t = xi11 · · · xinn ∈ Tn , we will use the symbol D(t) as D(i1 , . . . , in ).
Moreover, we write D = {D(t)|t ∈ Tn } and denote by SpanC (D) the C-vector space
generated by D. We now introduce some morphisms on D that act as ”derivative”
and ”integral”:
½
σxj (D(i1 , . . . , in )) =
D(i1 , . . . , ij − 1, . . . , in ) if ij > 0
0
otherwise
%xj (D(i1 , . . . , in )) = D(i1 , . . . , ij + 1, . . . , in )
(7.5)
(7.6)
Such operators extend trivially on SpanC (D) by linearity, and one can easily define
σt and %t for any t ∈ Tn by composition.
Definition 7.11. A subspace L of SpanC (D) is said to be closed if
σxj (L) ⊆ L, for all j = 1, . . . , n.
183
Definition 7.12. Let I be a primary ideal in R such that
√
I = (x1 , . . . , xn ). We
define the subspace of differential operators associated to I as
∆(I) := {L ∈ SpanC (D) | L(f )(0, . . . , 0) = 0 for all f ∈ I}.
Similarly, we associate to each subset V ⊆ SpanC (D) an ideal
I(V ) := {f ∈ R | L(f )(0, . . . , 0) = 0 for all L ∈ V }
Theorem 7.5. Let m be the maximal ideal (x1 , . . . , xn ) of R. There is a bijective
correspondence between m-primary ideals of R and closed subspaces of SpanC (D)
∆
{m − primary ideals in R} À{closed subspaces of SpanC (D)}
I
so that I = I∆(V ) and V = ∆I(I) for every I and V . Moreover, for a zerodimensional m-primary ideal of R whose multiplicity is µ, we have that dimC (∆(I)) = µ.
Theorem 7.5, whose complete statement and proof can be found in [55, 56], provides some properties of the space ∆(I) that are not obvious from its definition. For
example the noetherian operators associated to a zerodimensional primary ideal form
a closed subspace of SpanC (D). In addition, when considering a zerodimensional
primary ideal, it follows that the dimension of ∆(I) is finite, and so we can view a
basis of ∆(I) as a set of noetherian operators, that in this particular case happen to
be operators with constant coefficients. Moreover, such a vector space has the nice
property of being closed, fact that has been used by the authors of [56] to construct
a procedure that, given I, computes ∆(I). The algorithm is described below.
184
Algorithm 7.2. Let I be a zerodimensional primary ideal of R such that V (I) =
{(0, . . . , 0)}and let µ = dimC (R/I) be its multiplicity. The following procedure computes the noetherian operators associated to I:
Input: G = {g1 , . . . , gt } a Gröbner Basis for I.
Output: ∆(I) = {L0 , . . . , Lµ−1 }
Initialization: i = 1, L0 = 1 =IdSpanC (D)
If µ > 1, construct a linear operator L1 =
Pn
j=1 cj ∂xj
with an oppor-
tune choice of the cj ’s such that L1 (f )(0, . . . , 0) = 0 is satisfied for each
generator f of I. Put i = 2.
While i < µ do
define Li+1 as a linear combination of %j0 (L0 ), . . . , %ji (Li ) such
that
- hL0 , . . . , Li+1 i is closed and
- Li+1 (f )(0) = 0 for each generator f of I
Corollary 7.1. Let L be an operator of ∆(I), where I is as in algorithm 7.2 and µ
is its multiplicity. Then deg(L) < µ as an element of An .
Proof. The construction of ∆(I) starts with L0 = 1 and at each step the degree of
Li+1 increases of at most 1, so that the last element Lµ−1 has degree at most µ.
Remark 7.6. Algorithm 7.2 consists basically in the solution of a system of linear
equations in the coefficients cj of the linear combinations Li+1 = c0 %j0 L0 +· · ·+ci %ji Li .
185
Since the system can have more than one solution, the authors suggest to pick the one
with minimal leading term. An implementation for a version of 7.2 has been coded
for CoCoA and is available through the CoAlA web page [28].
Example 7.5. The following example is taken from [35] (p. 37, ex. 4). Here we
show how to study it using Algorithm 7.2. Let us consider the primary ideal at the
origin I = (y 2 , x2 − y) ⊂ C[x, y] whose multiplicity is 4. We start with L0 = 1
and an obvious choice for a linear operator is L1 = ∂x . This has also a geometric
interpretation: the origin is the intersection of the two curves given by the generators
y 2 (the x-axis twice) and x2 − y (a parabola with vertex at the origin). Such two
curves not only intersect at the origin but they are also tangent along the direction
of the x–axis, therefore L1 = ∂x must be a noetherian operator. The higher degree
operators describe a higher contact of the line and the parabola at zero. We can
try to find the next one as a combination L2 = a∂x + b∂xy. However, this operator
L2 does not respect the closure condition since σx (L2 ) = 1 + ∂y which is not in
the subspace hL0 , L1 i = h1, ∂xi. A different choice for the morphisms %xj , instead,
gives L2 = a∂y + b∂xy which respects closure and annihilates the generators of I
at zero with a = 1 and b =
1
.
2
Again, this operator could have been foreseen in
advance since it is the global annihilator of x2 − y and it trivially annihilates y 2 . As
a last operator, one can choose L3 = %x (L2 ) = ∂xy + 61 ∂x3 . Of course, the choice
%y (L2 ) = 21 ∂y 2 + 12 ∂x2 y would have been possible as far as the annihilation of I is
concerned, but it would have violated closure since σx (L2 ) = ∂xy is not a combination
of the previous operators. The iteration ends here since we have found 4 differential
operators.
186
7.4.2
An algorithm using the Taylor polynomial
We are now going to present an alternative procedure to compute the noetherian
operators associated to I that makes no use of linear algebra and utilizes the power
of Gröbner Bases .
Algorithm 7.3 (Computation of Noetherian Operators for Zerodimensional Ideals).
Let I be a zerodimensional primary ideal of R such that V (I) = {(0, . . . , 0)}. The
following procedure computes the noetherian operators associated to I:
Input: G = {g1 , . . . , gt } a Gröbner Basis for I.
Output: ∆(I) = {L1 , . . . , Lµ }.
- Compute µ(I) = dimC (R/I).
- Write the Taylor expansion at the origin of a polynomial h ∈ R
up to the degree µ − 1 with coefficients cα ∈ C:
|α|=µ−1
Tµ−1 h(x1 , . . . , xn ) =
X
cα xα1 1 . . . xαnn
|α|=0, α∈Nn
- Write the Normal Form of Tµ−1 h with respect to G as
NFTµ−1 h(x1 , . . . , xn ) =
X
dβ xβ1 1 . . . xβnn
β
and find scalars aβα ∈ C such that dβ =
P
α
aβα cα .
(7.7)
187
- For each β such that dβ 6= 0, return the operator
Lβ =
X
aβα
α
X
1
aβα D(α1 . . . αn ).
∂xα1 1 · · · ∂xαnn =
α1 ! · · · αn !
α
Proof. Let h(x1 , . . . , xn ) =
Pdeg(h)
α∈Nn
cα xα1 1 . . . xαnn be the Taylor expansion centered at
the origin of a polynomial h ∈ R and let G be the Gröbner Basis of I. From the
theory of Gröbner Bases we know that the normal form with respect to G of h is
zero if and only if h ∈ I, so the condition NF(h) = 0 is the one that we want to
characterize. It suffices to write
deg(h)
NF(
X
α∈Nn
deg(h)
cα xα1 1
. . . xαnn )
=
X
dβ xβ1 1 . . . xβnn = 0
(7.8)
β∈Nn
and deduce from the annihilation of each coefficient dβ in (7.8) a differential condition
on h. This completely characterize the membership of a polynomial h to I. The only
thing to observe is that we do not need to work with terms up to deg(h) for the Taylor
expansion. In fact, the number of differential conditions we need is precisely µ, and
so from corollary 7.1 it follows that the derivatives to be considered are, in the worst
case, the ones of order µ − 1. Those differential conditions arise by using coefficients
cα up to |α| = µ − 1. Therefore the Taylor expansion can be truncated at µ − 1.
Remark 7.7. It is crucial to observe that we do not need to characterize the membership of a polynomial h of undetermined degree deg(h) since we have the bound
µ − 1 on its degree. Thus Algorithm 7.3 is a procedure that is implementable on any
computer algebra software package. Moreover, the computation of the normal form
(7.7) can be done degree by degree, so that we can stop the computation whenever
188
the normal form of some degree is zero. This actually speeds up the computation of
the operators significantly (CPU times for several example are available at [28]).
Example 7.6. Consider again Example 7.5 to show the substantial difference between
procedures 7.2 and 7.3. Since µ(I) = 4 we start by writing the truncated Taylor
expansion of a polynomial h ∈ C[x, y]:
T3 h(x, y) = c00 + c10 x + c01 y + c20 x2 + c11 xy + c02 y 2 + c30 x3 + c21 x2 y + c12 xy 2 + c03 y 3
and perform the normal form computation using x2 → y and y 2 → 0 as rewrite rules.
Grouping like terms we can write the remainder of T3 h as a linear combination of the
generators 1, x, y, xy of R/I (the so called Macaulay basis for the ideal I, see again
[54], page 62) as follows:
NF(T3 h) = [c00 ] + [c10 ]x + [c01 + c20 ]y + [c11 + c30 ]xy
(7.9)
Note that the terms y 2 , x2 y, xy 2 and y 3 disappeared since they all rewrote to zero.
The computation ends by expressing the coefficients written into square brackets in
(7.9) as operators according to their meaning as Taylor coefficients. Namely [c00 ] →
1, [c10 ] → ∂x, [c01 + c20 ] → ∂y + 12 ∂x2 , [c11 + c30 ] → ∂xy + 16 ∂x3 . This gives the same
result obtained in the example 7.5 as expected. This is not surprising since theorem
7.5 states that the correspondence I ↔ ∆(I) is one-to-one.
Algorithm 7.3 does not take directly into account the closure of the space of
noetherian operators, as algorithm 7.2 did. The fact that ∆(I) is closed is a general
fact which follows from a ”Leibnitz formula” for the morphisms σxj and the fact that
I is an ideal (see [55], prop. 2.4). This is true not only for zerodimensional ideals
but also in higher dimension, as the closure holds even in higher dimensional cases
189
as we will see in section 9. We want to show that the closure of ∆(I) is also a direct
consequence of algorithm 7.3 and of the following property of Macaulay bases.
Lemma 7.1. Let I ⊂ R be an ideal and let M be the Macaulay basis of R/I, i.e.
the generators of R/I as a C-vector space. Let sxj : Tn → Tn be the ”derivative”
morphism
½
sj (xi11
· · · xinn )
=
i −1
xi11 · · · xjj
0
· · · xnin if ij > 0
otherwise
(7.10)
Then M is sj -closed for each j.
Proof. It is know that the Macaulay basis for R/I can be computed through a Gröbner
Basis G of I. In fact it is (see [54], theorem 1.5.7):
M = Tn \ LTσ (G)
where σ is any term ordering on Tn . Since G is a Gröbner Basis for I, the leading
term ideal LTσ (I) coincides with LTσ (G). Let t 6= 0 be a term of M. Suppose that
there exists an index j such that 0 6= sj (t) ∈
/ M. Then sj (t) ∈ LTσ (G). The latter
being an ideal, we have t = xj · sj (t) ∈ LTσ (G), which is a contradiction. Note that
if sj (t) ∈
/ M ∀ j, this simply says that t = 0 which is again a contradiction.
The morphism sj introduced in the above lemma is the analogue of σxj defined
in section 7.4.1, and we will show in the next proposition that the sj -closure of M is
equivalent to the σxj -closure of the space of noetherian operators associated to I.
Proposition 7.6. Let I be a zerodimensional primary ideal of R such that V (I) =
{(0, . . . , 0)} and let O = {Lβ } be the set of operators computed with algorithm 7.3.
Then SpanC ({Lβ }) is a closed subspace of SpanC (D).
190
Proof. Let Lβ ∈ O, and let dβ be the corresponding coefficient of the normal form
NF(h) as computed with the algorithm. Let xβ = xβ1 1 · · · xβnn be the term whose
coefficient is dβ . It is clear that such a term is part of the Macaulay basis of R/I
since it appears in the expression of NF(h), which is a representation of the class
of h in the quotient R/I. Denote by Fβ the set of operators of O such that the
corresponding term in the expression of NF(h) divides xβ :
Fβ = {Lγ ∈ O such that xγ |xβ }
and for each Lγ ∈ Fβ consider tγ = xβ−γ . Since each Lγ has been computed from the
Taylor expansion of using a division algorithm that uses a Gröbner Basis G of I, we
have that (see [54], prop. 2.2.2) if h0 is such that
xγ = NF(h0 ) and supp(h0 ) ⊆ supp(h)
then
xβ = tγ xγ = NF(tγ ) NF(h0 ) = NF(tγ h0 )
i.e. the term in xβ is obtained rewriting a multiple of that part of the polynomial h
which rewrites to xγ . By looking at the expression of Lβ is then obvious that
σtγ (Lβ ) = Lγ
since Lβ is written as a combination of Taylor coefficients corresponding to the terms
of tγ h0 . It now suffices to prove that such tγ ’s are enough to conclude that O is closed.
This is a consequence of the previous lemma, since all the dγ in Fβ are associated to
those terms xγ of the Macaulay basis M that divide xβ , hence from the sj -closure of
191
M we deduce that {xγ = stγ (xβ )} = {sj (xβ ), j = 1, . . . , n}.
7.5
Extension to modules
In this section we explore the possibility of extending algorithm 7.3 to zerodimensional
primary modules. It is mostly a matter of notation, since all the results and main
tools we used in the previous section hold naturally for modules. Recall that for
submodules M of Rs , s ∈ N, an analogue of the primary decomposition described
in section 7.2 holds. In fact M is a submodule of a noetherian module, hence it is
finitely generated.
7.5.1
Multiplicity variety for modules
The first step we need to make is to understand how a primary decomposition allows
the definition and the computation of a multiplicity variety.
Definition 7.13. Let M be a submodule of Rs , s ∈ N and let A be the r × s matrix
of polynomials whose rows generate the module M . Let JM be the annihilator of M
in Rs ,
JM = Rs :R M = 0 :sR /M
The characteristic variety V (M ) associated to M , is the (algebraic) set of common
zeros of the maximal minors of A, or equivalently, the variety V (JM ).
Therefore, the geometric object attached to a module is V (M ) = {P ∈ Cn | rk(A) <
s at P }. Of course, when s = 1 and M is an ideal I, such a variety coincides with
the algebraic set V (I). Let us introduce the concept of primary module through the
following definition:
192
Definition 7.14. The radical of a module M ⊆ Rs is the ideal
√
M = {f ∈ R | f t M ⊆ M for some t ∈ N} =
p
JM ,
and M is said to be primary if
∀f ∈ R and v ∈ Rs such that f · v ∈ M, either v ∈ M or f ∈
√
M.
Proposition 7.7. Let M be a submodule of Rs . Then there exists a collection
Q1 , . . . , Qt of (distinct) primary submodules of Rs such that
M = Q1 ∩ · · · ∩ Qt and JM = JQ1 ∩ · · · ∩ JQt .
In particular, V (M ) = V (Q1 ) ∪ · · · ∪ V (Qt ).
In order to introduce a multiplicity variety for M , we still need to define the space
of operators acting on vectors of Rs .
Definition 7.15. A differential operator with polynomial coefficients acting on Rs is
P
an element ∂ = si=1 ∂i ei , ∂i ∈ An , ei being an element of the canonical basis of Rs ,
such that
s
s
X
X
∂(
fi ei ) =
∂i (fi )ei .
i=1
i=1
Denoting with Ds the set of elementary differential operators of the form D(t)ei ,
t ∈ Tn , the space of all differential operators will be then SpanC (Ds ).
Definition 7.16. Let V1 , . . . Vt be algebraic sets in Cn and let ∂1 , . . . , ∂t be differential
operators in SpanC (Ds ). We say that the set
V = {(V1 , ∂1 ); (V2 , ∂2 ); . . . ; (Vt , ∂t )}
193
is a multiplicity variety. Given a submodule M of Rs , V is said to be associated to
M if v ∈ M ⇔ ∂ j (v)|Vj = 0, for all j. In this case the operators ∂j ’s are called
noetherian operators associated to M .
7.5.2
Noetherian operators for a zerodimensional module
As we did for ideals, we now discuss the explicit construction methods of the noetherian
operators associated to a zerodimensional module.
Definition 7.17. A submodule M of Rs will be called zerodimensional if the Krull
dimension of Rs /M is zero or, equivalently, if dimC (Rs /M ) is finite. The vector space
Rs /M is then generated by a finite number of module terms of type tei (where t ∈ Tn
and ei is an element of the canonical basis of Rs ) which form the Macaulay basis of
Rs /M .
The cardinality of the Macaulay basis for M corresponds obviously to µ(M ) and
will be again a key tool in the computation of the ∂ j . Let us generalize (7.5) and
(7.6) to the case of SpanC (Ds ):
Definition 7.18. Let D(i1 , . . . , in )ek ∈ Ds be an elementary differential operator.
Define
½
σxj (D(i1 , . . . , in )ek ) =
D(i1 , . . . , ij − 1, . . . , in )ek if ij > 0
0
otherwise
(7.11)
and
%xj (D(i1 , . . . , in )ek ) = D(i1 , . . . , ij + 1, . . . , in )ek .
(7.12)
Again we can extend those morphisms by linearity on the whole space SpanC (Ds ) and
define by composition σt and %t for each term t ∈ Tn . The closure of a subspace L of
194
SpanC (Ds ) is again defined by the condition
σxj (L) ⊆ L, ∀j.
Given a module M in Rs and a subspace V of SpanC (Ds ), it is possible to define
∆(M ) ⊆ SpanC (Ds ) and M(V ) ⊆ Rs
similarly to what we did for ideals.
Remark 7.8. It is very important for our purposes to note that even in the case of
primary modules, the space of operators ∆(M ) is closed. In fact, the proof of this
fact for ideals (see [55], prop. 2.4) only uses a Leibnitz formula for σxj , which still
holds given that (7.11) is formally identical to (7.5), and the fact that M is a module.
Thus, the proof remains formally the same even in the case of modules.
We will now state a procedure, similar to algorithm 7.3, from which it will follow
that not only the computation of noetherian operators is possible for zerodimensional
modules, but it will also provide a new proof of the fact that a multiplicity variety of
such modules exists, things that could not be derived from the one-to-one correspondence 7.5 since this fact has been showed only for ideals. Again, we will assume that
the module is primary and centered in zero.
Algorithm 7.4 (Noetherian Operators for Zerodimensional Modules). Let M be a
zerodimensional primary submodule of Rs such that V (M ) = {(0, . . . , 0)}. Moreover,
let σ be any module term ordering on Tn he1 , . . . , es i. The following procedure computes the noetherian operators associated to M :
Input: G = {v1 , . . . , vt } a Gröbner Basis for M .
195
Output: {Ljβ }, a set of noetherian operators associated to M .
- Compute µ(I) = dimC (Rs /I).
- Write the Taylor expansion at the origin of a vector w ∈ M
up to the degree µ − 1 with coefficients ciα ∈ C:
Tµ−1 w(x1 , . . . , xn ) =
s
X
|α|=µ−1
X
ciα xα1 1 . . . xαnn ei
i=1 α=0 α∈Nn
- Write the Normal Form of Tµ−1 w with respect to G as
NFTµ−1 w(x1 , . . . , xn ) =
s X
X
i=1
j
and find scalars aji
βα ∈ C such that dβ =
diβ xβ1 1 . . . xβnn ei
(7.13)
β
P
α, i
i
aji
βα cα .
- For each pair (β, j) such that djβ 6= 0, return the operator
Lβ =
X
α, i
aji
βα
X
1
∂xα1 1 · · · ∂xαnn =
aβα D(α1 . . . αn )ei .
α1 ! · · · αn !
α, i
Proof. The proof is formally identical to the one given in the case of an ideal I. Note
that closure of the space of operators guarantees that the degree of the operators is at
most µ(M ) − 1, which allows to truncate the Taylor expansion at the degree µ − 1. As
for the rest of the procedure, the properties of Gröbner Bases and normal forms for
modules are identical to the ones for ideals, so that the annihilation of the coefficients
of the normal form provides a membership test.
196
Remark 7.9. The closure property of the space of operators given by the previous
procedure is again a direct and independent result. Any set of differential operators
computed as above ends up to be closed since the Macaulay basis is sxk –closed,
provided that the definition of sj given in (7.10) is extended to modules.
Example 7.7. Let A be the matrix


x 1
A= y x 
0 y
and let M be the module generated by the rows of A, i.e. M = hxe1 + e2 , xe2 +
ye1 , ye2 i. The module term ordering we choose is ToPos with σ = Lex, meaning that
to compare two terms we first look at the power product, using Lex, and then we
look at the position. The way we just wrote the generators of M reflects this choice.
It is clear that JM = (x2 − y, y 2 , xy), and, using for example CoCoA, we find out that:
- µ(M ) = 3
- the Lex–Gröbner Basis of M is G = {xe1 + e2 , xe2 + ye1 , ye2 , y 2 e1 }
- a Macaulay basis for M is the set {e1 , e2 , ye1 }.
We begin by writing explicitly the vectorial Taylor expansion of a vector w(x, y) ∈ Rs
up to degree 2:
T2 w(x, y) = c100 e1 + c200 e2 + c110 xe1 + c210 xe2 + c101 ye1 + c201 ye2 + c120 x2 e1 + c220 x2 e2 +
c111 xye1 + c211 xye2 + c102 y 2 e1 + c202 y 2 e2 .
Only few terms survive after we compute the normal form relative to the Gröbner
Basis G, leading to
NF(w) = [c100 ]e1 + [c200 − c110 ]e2 + [c120 + c101 − c210 ]ye1 .
197
We conclude that the noetherian operators associated to M , written in vectorial form,
are
1
D00
= (id, 0),
2
D00
= (−∂x, id),
1
1
D01
= ( ∂x2 + ∂y, −∂x)
2
2
1
and it is easy to check that they generate a closed subspace since σx (D00
) = σy (D01
)=
2
1
1
.
) = D00
and σx (D01
D00
Example 7.8 (Solution of a System of PDE’s). In Section 7.2 we saw that the
Fundamental Principle can be used to write an integral representation of the solution
of a system of linear constant coefficient partial differential equations. We will show
how this can be applied, now that we know how to compute noetherian operators.
Consider the overdetermined PDE system given by

fzz − fz + ft + 2gz



fzt + gt
ftt + gzt − gt



ft − gzz + gz + gt
=
=
=
=
g
0
0
0
(7.14)
where f, g ∈ C ∞ (R2 ) and we use indices to denote derivatives. The general solution
to (7.14) can be written using a generalization of (7.4). We consider the rectangular
operator P (D) defined by


x2 − x + y
2x − 1


xy
y

P =

y2
xy − y 
y
x2 − x − y
where x and y are the dual variables of z and t respectively. Note that we are choosing
a particular Fourier Transform to write P (D) so that it does not take into account
√
the factor − −1. The module M associated to the matrix P is not primary, hence
198
we can use Singular to get a primary decomposition (using the function modDec form
the library mprimdec.lib). M is the intersection of the two zerodimensional modules
M1 = h(x, 1), (y, x), (0, y)i,
J1 =
√
M2 = h(x − 1, 1), (y, 0), (y, x − 1)i,
M1 = (x, y)
J2 =
√
M2 = (x − 1, y)
of multiplicity, respectively, 3 and 2. We already computed the operators associated
to the module M1 in the previous example. To compute the operators associated to
M2 we need to shift the variety to the origin using the change of coordinates (X =
x − 1, Y = y). Then, using the new variables X and Y , we can apply Algorithm 7.4
and find the noetherian operators: {(id, 0), (∂X, −id)}. Going back to the variables
x, y we have the set {(id, 0), (∂x, −id)}. Therefore, it is possible to write explicitly
the solutions to (7.14) as
µ
f (z, t)
g(z, t)
µ
+D
7.6
id
0
¶
µ
=A
id
0
¶
µ
zx+ty
e
|(0,0)
¶
µ
zx+ty
e
|(1,0)
+B
+E
∂x
−id
−∂x
id
¶
¶
e
e
=
+C
|(0,0)
µ
zx+ty
|(1,0)
µ
zx+ty
1
∂x2
2
+ ∂y
−∂x
¶
ezx+ty +
|(0,0)
A − Bz + 12 Cz 2 + Ct + Dez + Ezez
B − Cz − Eez
The positive dimensional case
When dealing with ideals and modules whose Krull dimension is higher than zero
one may expect that the fact that the associated noetherian operators are constant
coefficient linear operators does not hold anymore. In fact, this is the case for some of
the examples from the literature (see [35,60]). For example, when considering the ideal
I = (x2 , y 2 , −xz + y) ⊂ C[x, y, z] one has that a set of noetherian operators associated
¶
199
to I is {1, ∂x+z∂y} and it can be proved that there exist no set of noetherian operators
with constant coefficients associated to I (see [60], example 4, p. 183). However, an
interesting property that we notice in this case is that the set of variables with respect
to which the derivatives are taken is disjoint from the set of variables appearing in the
polynomial coefficients (in this case such sets are respectively {x, y} and {z}). This is
actually valid whenever we can put the variety associated to the ideal in a particular
position, through an opportune change of coordinates, called normal position. To
do this, one can apply the procedure of Noether normalization to the ideal I. This
algorithm comes from the so-called Noether Normalization Theorem (see [8], p. 116).
We now state a version of the theorem that we will need for our computations:
Theorem 7.6 (Noether Normalization Theorem). Let I be a primary ideal of C[z1 , . . . , zn ].
There exist a non–negative integer d and a (linear) change of coordinates
ϕ : C[z1 , . . . , zn ] → C[x1 , . . . , xn−d , t1 , . . . , td ]
such that:
a) ϕ(I) ∩ C[t1 , . . . , td ] = (0),
b) C[z1 , . . . , zn ]/I is a finitely generated C[t1 , . . . , td ]–module,
c) for each i = 1 . . . n − d, ϕ(I) contains a polynomial of the form
Qi (t1 , . . . , td , xi ) = xei i + p1 (t1 , . . . , td )xei i −1 + · · · + pei (t1 , . . . , td )
where ei is the degree of the polynomial Qi .
The ideal ϕ(I) is said to be in normal position with respect to the variables x1 , . . . , xn−d .
Remark 7.10. The proof of the Normalization Theorem can be found for example
in [8], in the case of prime ideals. However, as shown in [45], the result holds for
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the general case with the exception of condition a) which requires that I be primary.
If the ideal I is prime, the polynomials Qi in condition c) can be chosen to be irreducible. The proof of the theorem provides an algorithm to achieve the normal
position. Basically, one constructs the polynomial Qi at each step, performing a coordinate change such that Qi has a monic leading term of the form xei i and then
eliminating the variable xi . The coordinate change used at each step is generic. The
procedure to compute the Noether normalization of an ideal has been studied in [52]
and it is available in Singular in the library algebra.lib (see [46]). We coded a
version of the algorithm for CoCoA as well, [28].
Theorem 7.6 basically states that it is possible to find a new system of coordinates
where the x variables act as ”variables” and the t variables act as ”coordinates”, and
where the integer d appearing in 7.6 is nothing but the dimension of the ideal I.
Hence, if we make the variables t invertible, i.e. if we extend the ideal to the ring
C(t)[x] where C(t) is the ring of quotients of C[t], we end up with a zerodimensional
ideal. Furthermore, since we are interested in primary ideals, we may expect that the
extension of the ideal to C(t)[x] is still primary. The following proposition assures
that such facts hold if I is in normal position.
Proposition 7.8. Let I = (f1 , . . . , fr ) be a primary ideal of dimension d in the
polynomial ring R = C[x1 , . . . , xn−d , t1 , . . . , td ], in normal position with respect to
x1 , . . . , xn−d . Denote by Rd = C(t1 , . . . , td )[x1 , . . . , xn−d ] the ring of polynomials in the
x variables with coefficients in the field of fractions C(t1 , . . . , td ) = Frac(C[t1 , . . . , td ]).
The following facts hold:
1) the inclusion map ϕ|I : I ,→ IRd is injective and IRd ∩ R = I,
2) the extended ideal IRd is primary,
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3) the extended ideal IRd is zerodimensional.
Proof. The fact that the inclusion is injective is trivial. To prove 1), let us consider
a polynomial f in R ∩ IRd . As an element of IRd it can be written in the form
f=
r
X
ai (x, t)
i=1
bi (t)
fi (x, t)
where x = (x1 , . . . , xn−d ), t = (t1 , . . . , td ), and ai and bi are just polynomials in the set
of variables indicated in parenthesis. Let b(t) = b1 (t) · · · br (t) and consider the product
bf . Both b and f are polynomials in R and their product is an R-linear combination
of the generators of I, so bf ∈ I. Since I is primary it follows that either bm ∈ I
for some positive integer m or f ∈ I. The first possibility is in contradiction with
condition a) of the Noether normalization, hence f ∈ I. This proves that IRd ∩R ⊆ I.
The opposite inclusion is trivial, so we conclude that IRd ∩ R = I. The same type of
argument can be used to prove that IRd is primary: consider two fractions
f (x, t) =
a(x, t)
,
b(t)
g(x, t) =
c(x, t)
d(t)
such that f g ∈ IRd . Then (bf ) · (dg) is a polynomial in I and since I is primary we
either have bf ∈ I or dm g m ∈ I for some positive integer m. In the first case, using
again that I is primary and using condition a) of Theorem 7.6, we get that f is in I.
In the second case we have that g m is in I. Therefore either f ∈ IRd or g m ∈ IRd .
Finally, statement 3) follows from the general theory of the Krull dimension, since
(t¯1 , . . . , t¯d ) is a maximal regular sequence in R/I that reduces to just constants when
extending the ideal to Rd .
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Remark 7.11. The fact that a normalization process is necessary in order to compute
the noetherian operators has been pointed out in [59] and does not surprise us since
its tight connection to primary decomposition. Once the ideal has been put in normal
position, one can calculate the noetherian operators using the procedure used for the
zerodimensional case applied to the ideal IRd . In order to make this possible, there
is a further assumption to make on the ideal I. We need the characteristic variety
of the extended ideal IRd to be the origin (0, . . . , 0) in C(t)d . This is necessary if
one wants to use the Taylor expansion at zero of a polynomial h(x) ∈ C(t)[x]. In
section 7.4 we worked with C as ground field, and since it is algebraically closed,
we were able to use the fact that every primary ideal is associated to a single point
(with multiplicity). It was then just a matter of a C–linear change of coordinates for
this point to be the origin. Instead, since C(t) it is not algebraically closed, we have
that primary ideals can be associated to conjugate roots that are in a finite algebraic
extension of C(t). Consider, for example, the primary ideal I = (x2 − t) in C[x, t]. It
is obviously in normal position with respect to x and its dimension is 1. If we extend
I to C(t)[x] we immediately see that the ideal IC(t)[x] is still zerodimensional and
primary, but its characteristic variety is empty. If we introduce the new symbol a such
that a2 = t, then IC(t)[a][x] is not primary anymore since (x2 − t) = (x − a) ∩ (x + a)
in the new ring. Only at this point one could apply a technique like Algorithm 7.3
to each primary component, performing a change of coordinates like x → x ± a to
shift the point to the origin. This makes the application of the algorithm not quite
as immediate as it may seem, since both the primary decomposition and the change
of coordinates make use of coefficients in C(t)[a].
Before we move on and present an equivalent version of Algorithm 7.3 for positive
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dimensional ideals, there is still one more step. Formerly, when treating the zerodimensional case, we chose to start with a Gröbner Basis for the ideal I, computed
with respect to any term ordering. This is no longer possible if we want to extend the
procedure to the higher dimensional case. In fact, after we perform the normalization,
the variables t play some different role in the recipe, as they serve as ”constants” once
we extend I to Rd = C(t)[x].
Example 7.9. Consider the ideal I = (x2 − t, xt − 1) in C[x, t]. A DegLex–Gröbner
Basis for I (with x > t) is given by G = {x2 −t, xt−1, t2 −x}, where the leading term
are highlighted in bold. When we look at such polynomials in Rd , however, we see
that the leading terms change, in fact the last polynomial should better be written
as −x + t2 . Note that in this case the extended ideal IRd happens to be the whole
ring Rd since the polynomial t3 − 1 belongs to IRd , and such polynomial is a constant
in C(t)[x]. It is a necessary and sufficient condition for an ideal to be the whole ring
that any Gröbner Basis with respect to any ordering contains a constant polynomial.
If we look at G we see that there is no such a constant, meaning a polynomial only in
the variable t, and hence we conclude that the set G does not form a Gröbner Basis
for IRd , with respect to the ordering DegLex restricted to the terms in x. If we choose
instead the term ordering Lex, a Gröbner Basis for I is given by G = {−x+t2 , t3 −1},
and in this case we do have a polynomial in t appearing, making G a Gröbner Basis
for IRd .
As the example shows, we really want the variables x to be the ”main” variables
with respect to which the Gröbner Basis needs to be computed. This can be achieved
using Lex, but more generally using elimination. We now introduce some definitions
about elimination theory and term orderings (see [54], section 3.4, for details on this
204
topic).
Definition 7.19. Let R = C[x, t] where x = (x1 , . . . , xn−d ), t = (t1 , . . . , td ). A term
ordering σ on Tn is called an elimination ordering with respect to x if every element
f ∈ R whose leading term is contained in C[t] is such that f ∈ C[t]. In other words,
∀f ∈ R,
LTσ (f ) ∈ C[t] ⇒ f ∈ C[t].
The reason why such a term ordering is called an elimination ordering is that it
allows to eliminate the variables x from an ideal, i.e. it allows to compute I ∩ C[t].
To do this, it suffices to compute a Gröbner Basis with respect to any elimination
ordering as in definition 7.19 and then keep only the elements that do not contain any
monomials in x. Such elements actually form a Gröbner Basis for the ideal I ∩ C[t].
It can be easily checked that Lex is an elimination ordering with respect to any
”initial” subset of variables, i.e. with respect to any subset of the type {x1 , . . . , xk }
in C[x1 , . . . , xn ], with k ≤ n. A class of term orderings that satisfy the elimination
property and that we are going to use for our goal of computing the noetherian
operators in C(t)[x] are the so called product orderings.
Definition 7.20. Let R = C[x, t] as before and let σx and σt be two term orderings
on the set of terms Tx = {xa | a ∈ Nn−d } and Tt = {tb | b ∈ Nd } respectively. The
product order σx · σt is defined by
xa tb >σx ·σt xc td ⇔ xa >σx xc or (xa = xc and tb >σt td ).
It is immediate to show that the product ordering defined above is an elimination
ordering with respect to x, no matter what the choice of σx and σt is. Elimination
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orderings are usually slow when it comes to Gröbner Basis computations, in particularly Lex is known to be one of the slowest. Product orderings are then introduced
to perform better. One can in fact define a ”fast” term ordering (such as DegRevLex)
on each of the two subsets of variables, and then take the product. The following
lemma answers the question we posed above.
Lemma 7.2. Let R = C[x, t] be a polynomial ring equipped with a product ordering σ
of the type σx · σt as in definition 7.20. Let I be an ideal of R and let G = (g1 , . . . , gs )
be a σ–Gröbner Basis for I. Consider the extended ideal IRd in Rd = C(t)[x] endowed
with the term ordering σx . Then G forms a Gröbner Basis for IRd with respect to σx .
Proof. denote by xai tci the leading term of gi , where ai ∈ Nn−d and ci ∈ Nd , i =
1, . . . , s. From the fact that we chose a product ordering σ, it follows that once we
view gi as an element of IRd , its leading term is xai . In other words, LTσx (gi ) = xai
in Rd . Consider a polynomial f in IRd . The set G still forms a set of generators
for the extended ideal, so f can be written as an Rd -linear combination of the gi ’s.
Moreover, supposing f monic, we can write f as
f = xa +
X
pb (t)xb , where b ∈ Nn−d and xa >σx xb ∀ b.
b
Consider the product D(t) of all the denominators of the coefficients pb (t) in f . Then
D(t)f is a polynomial in R and it is still a combination of the elements of G, so
D(t)f ∈ I. Because of the fact that σ is a product order, the leading term of D(t)f
is simply the leading term of f multiplied by some power of t, i.e. LTσ (D(t)f ) = xa tc
for some c ∈ Nd . Hence, G being a Gröbner Basis for I, xa tc is a multiple of one of
the leading terms of its elements, say xa1 tc1 modulo a change on the order in G. This
206
means that there exist α ∈ Nn−d and γ ∈ Nd such that
xa tc = xα tγ xa1 tc1
which means that xa is a multiple of xa1 , and this concludes the proof.
We now have all the ingredients to generalize Algorithm 7.3 to the case of an ideal
of dimension greater than zero. As in section 7.4, we will suppose that a primary
decomposition of the ideal has already been calculated.
Algorithm 7.5 (Noetherian Operators for Non-Zerodimensional Ideals). Let d be a
positive integer, x = (x1 , . . . , xn−d ) and t = (t1 , . . . , td ) be variables and let σ = σx · σt
be a product ordering . Let I be a primary ideal in R = C[x, t]. Suppose that I
is in normal position with respect to x. Moreover, let IRd be the extended ideal in
Rd = C(t)[x] and suppose that the characteristic variety of IRd in C(t)d is the origin.
The following procedure computes the noetherian operators associated to I:
Input: G = {g1 , . . . , gr } a σ–Gröbner Basis for I.
Output: a set of noetherian operators for I.
- Compute the multiplicity of the ideal, µ(I).
- Write the Taylor expansion at the origin of a polynomial h ∈
C[x]
up to the degree µ − 1 with variable coefficients cα :
|α|<µ
ĥ := Tµ−1 h(x1 , . . . , xn−d ) =
X
α∈Nn−d
α
n−d
cα xα1 1 . . . xn−d
(7.15)
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- Let xai tbi be the leading term of gi and define tγ := tb1 · · · tbr
Repeat
- Multiply ĥ by tγ and compute its normal form with
respect to G. Rename that as ĥ:
ĥ := NF(tγ ĥ) =
X
β
n−d
dβ (t)xβ1 1 . . . xn−d
(7.16)
β
Until the number of nonzero dβ is exactly µ.
- For each β such that dβ 6= 0, find polynomials aβα (t) such that
P
dβ (t) = α aβα (t)cα and return the operator
Lβ =
X
α
aβα (t)
X
1
αn−d
=
aβα (t)D(α1 , . . . , αn−d , 0, . . . , 0).
∂xα1 1 · · · ∂xn−d
α1 ! · · · αn−d !
α
Proof. Let h be a polynomial of R. We want to characterize the membership of h to
I. Since we are assuming I in normal position, by condition 1) of Proposition 7.8 this
is equivalent to the membership of h to IRd . Since the latter is a zerodimensional
ideal of multiplicity µ, h ∈ IRd if and only if the Taylor polynomial of degree µ − 1
of h, with coefficients in C(t) reduces to zero when rewriting it using a Gröbner Basis
for IRd . This follows from the same proof of the algorithm 7.3 for zerodimensional
ideals. By Lemma 7.2, a σx –Gröbner Basis for IRd is given by the same elements
of the Gröbner Basis of I. Therefore, computing a normal form in I and in IRd
is equivalent. However, when writing the Taylor expansion as in (7.15), we need to
consider that the coefficients cα also depend on t. In order to be able to perform
a one-step reduction, we need each term in (7.15) to be at least multiplied by tγ .
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This does not affect Tµ−1 h as a polynomial in Rd since it is just a multiplication by
a constant. Also when considering the expression (7.15) in C[x, t], the effect of the
multiplication does not change the annihilation of NF(Tµ−1 h), since obviously
NF(Tµ−1 h) = 0 ⇔ NF(tγ Tµ−1 h) = 0.
The one-step reduction is then iterated enough times in (7.16) until we reach a sufficiently small number of nonzero terms (namely µ). By what we have proved so far,
it is then clear that at the end of the process the polynomial ĥ is exactly the normal
form of Tµ−1 h as a polynomial in Rd and hence the annihilation of its coefficients is
equivalent to the condition h ∈ IRd .
Remark 7.12. The main difference with respect to the algorithm for zerodimensional
ideals is that in this case we do not know if after just one step of reduction we have
achieved the normal form of the polynomial h(x, t), since the multiplication by tγ
could not be enough to assure that h has been rewritten to a sum that runs over just
the Macaulay basis terms for IRd . Multiplying Tµ−1 h once by tγ is definitely enough
for a one-step reduction of each term of the Taylor expansion. That is, each term is
being rewritten using at most one of the elements of the Gröbner Basis. However,
further reductions might occur if we multiply again by tγ . Also, note that such an
iteration has to terminate because σx is a well ordering.
Remark 7.13. The reduction step (7.16) for ideals with few generators is not very
heavy, but performing it multiple times could slow down the procedure by a significant
amount. We believe that it is possible to find an exponent γ1 large enough so that
we need to multiply by tγ1 just once, allowing the reduction to bring ĥ all the way
209
down to its final expression. For example, choosing γ1 = µ · γ seems to work fine at
least in the cases we tested, without the need of an iteration.
When applying Algorithm 7.5 to an ideal I in normal position, some redundant
factors in t could appear as an effect of the iterative multiplication by tγ at each step.
Since such factors are constants in Rd , they are actually not needed to characterize
the membership of a polynomial in Rd . It is then possible to eliminate these factors
from the final expression of the noetherian operators. The next example will clarify
what we mean.
Example 7.10. Consider the system of partial differential equations in three variables
given by

 fxx = 0
fyy = 0 .

fy = fxt
Its solutions are differentiable functions of the form f (x, y, t) = A(t) + B(t)x + B 0 (t)y,
where A and B are arbitrary functions of t. We want to derive this last statement
using the fundamental principle. The primary ideal associated to the system is I =
(x2 , y 2 , −xt+y) in C[x, y, t] (see [60]). If we consider the Lex ordering where x > y > t,
a Gröbner Basis for I is given by (x2 , xy, y 2 , −xt + y). Let us compute the associated
noetherian operators using Algorithm 7.5. It is immediate to check that I is in normal
position with respect to x and y and that, after inverting t, the variety associated
to IC(t)[x, y] is the origin in C(t)2 . The multiplicity of I can be computed with
CoCoA , and it is µ = 2. So we just need to write a linear polynomial h with variable
coefficients and multiply it by t, which is the only term in t appearing in the leading
210
terms of the Gröbner Basis:
T1 ĥ = T1 th = tc00 + tc10 x + tc01 y.
The only rewrite rule that we need to use to reduce h is hence xt → y which leads to
the final expression for the normal form
NF(ĥ) = [tc00 ] + [c10 + tc01 ]y.
Since the terms in x and y of the last expression are exactly µ = 2, we do not need
to proceed further and then we conclude that the operators are {t, ∂x + t∂y}. Since
the first is a multiple of t, we can divide it by t and get the final set {1, ∂x + t∂y}.
Now we can write the integral formula for the general solution of the system, using
ζ, η, τ as dual variables:
Z
Z
i(xζ+yη+tτ )
f (x, y, t) =
e
ζ=η=0
Z
(∂ζ + τ ∂η)ei(xζ+yη+tτ ) dµ2 (ζ, η, τ ) =
dµ1 (ζ, η, τ ) +
ζ=η=0
Z
Z
Z
Z
itτ
itτ
itτ
e dµ1 (τ )+ i(x+yτ )e dµ2 (τ ) =
e dµ1 (τ )+x ie dµ2 (τ )+y iτ eitτ dµ2 (τ ).
itτ
=
R
R
R
R
R
The last expression gives exactly the general solution as anticipated above. One just
has to consider arbitrary Radon measures dµ1 (τ ) = Â(τ )dτ and dµ2 (τ ) = B̂(τ )dτ
where  and B̂ are the Fourier transforms of the two arbitrary functions A and B.
7.7
An optimization of the algorithm
Let us go back to Example 7.6. The zerodimensional ideal I = (x2 − y, y 2 ) ⊂
C[x, y] has multiplicity 4 and the associated noetherian operators are {1, ∂x, ∂y +
211
1
∂x2 , ∂x∂y
2
+ 16 ∂x3 }. If we consider the monomial ideal LT(I) = (x2 , y 2 ) we see that
the noetherian operators are the monomial differential operators
{1, ∂x, ∂y, ∂x∂y}
(7.17)
obtained from the residual monomials of C[x, y]/I = h1̄, x̄, ȳ, xyi
¯ by just taking their
Fourier transforms. As we can see, the operators (7.17) constitute the ”initial” part
of the operators associated to I, i.e. they are the terms of minimal degree. We can
state the following result
Proposition 7.9. Let I be a zerodimensional ideal of R centered at the origin. Then
the number of noetherian operators associated to I and to LT(I) is the same. Moreover, the operators associated to LT(I) are the initial monomials of the operators
associated to I.
Proof. The first fact is a consequence of a theorem of Macaulay (see [54]) which
states that the vectors spaces R/I and R/ LT(I) are isomorphic. To prove the second statement let us consider the technique used in Algorithm 7.3 to construct the
noetherian operators. The key part of the algorithm is to write the Taylor expansion
of a polynomial up to degree µ−1 where µ is the multiplicity if I. We then reduce the
Taylor polynomial to its normal form using a Gröbner Basis of I. Such normal form
is a combination of the residual monomials (the Macaulay basis of R/I) with some
generic coefficients. Each residual monomial m has, at this point of the construction, a coefficient given by all the Taylor coefficients of the monomials that have been
rewritten to m. Such coefficient is somehow the ”memory” of the rewrite steps that
have lead to some monomials of the Taylor expansion to rewrite m. In particular the
Taylor coefficient of m, which when reinterpreted as a derivative is nothing but D(m)
212
using the notation of the previous section, is still part of the coefficient of m in the
normal form. Therefore, we conclude that for each residual monomial m, the term
D(m) always appears in one noetherian operator and since the residual monomials
are minimal, D(m) is minimal as well.
An alternative to Algorithm 7.3. We could think of performing the reduction
step of the algorithm for the computation of noetherian operators for zerodimensional
ideals ”backwards”. Instead of writing the full Taylor expansion and then using the
Gröbner Basis of I to rewrite it, we start from the residual monomials, which are
easily calculated for example with CoCoA. We then ”pull back” each monomial using
the generators of I as ”anti-rewrite rules”. Let us explain what we mean by this.
In general, when using a polynomial f to rewire another polynomial g, we use its
leading monomial LT(f ) to divide the polynomials g and then we substitute each
LT(f ) in g with the tail of f , LT(f ) − f . For instance, we rewrite g = x3 to xy using
f = x2 − y, by replacing x2 in x3 with the tail x2 − (x2 − y) = y. This operation,
when performed using the elements of a Gröbner Basis for I does not alter the class
of g in R/I and leads to the normal form NF(g). What we mean by ”anti-rewriting”
is, roughly speaking, to use the smallest monomial of f , in(f ), and replace it with
the head of the polynomial, in(f ) − f . This way, from in(f ) we ”climb up” to find all
the other monomials that are equivalent to in(f ) modulo (f ). Here is a more precise
definition.
Definition 7.21. Let f be a polynomials of R, let g be a monomial and let m = in(f )
be the smallest term of f with respect to a given term ordering on Tn . We say that
213
g rewrites backwards to g 0 in one step, using f , if m divides g and
g0 =
g
(m − f ).
m
Example 7.11. With this terminology, g = xy rewrites backwards to x3 using x2 −y,
which is exactly the opposite of the standard rewrite process that leads from x3 to
xy. If we use f = x2 + xy − 2y instead, g = xy rewrites to g 0 = 12 x3 + 21 x2 y. Finally,
g could not be rewritten backwards suing x2 − y 2 since y 2 does not divide g. Notice
that in general if we perform a one-step backwards reduction and then a one-step
reduction in the usual way, we obtain back g.
We can now apply an iteration of this procedure of rewriting backwards a monomial using a Gröbner Basis for I. We start from a residual monomial and we rewrite it
backwards using one generator. Then we rewrite backwards each monomial obtained
after this step, if possible, using any element of the Gröbner Basis. Technically this
procedure never ends as we can imagine to obtain a nw polynomial of higher degree
at each step, like when rewriting g = x using f = x2 − x. However, for the purpose
of computing noetherian operators, we know from section 7.4 that, as polynomials in
C[∂x1 , . . . , ∂xn ], they have degree at most µ − 1. Therefore we can stop the iteration
once we have reached a polynomial of such degree. Let us illustrate this ida with an
example before we present the algorithm in general.
Example 7.12. Consider the ideal J = (x2 − z, y 2 − z, z 2 ) in C[x, y, z]. It represents
the origin in C3 with multiplicity eight. Its generators are a DegLex Gröbner Basis.
The residual monomials for R/J are
{1, x, y, xy, z, xz, yz, xyz}.
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First, let us reconstruct the noetherian operators associated to xyz. By rewriting it
using x2 − z we obtain the new monomial x3 y. This cannot be rewritten further.
However, the term xy 3 is another monomial that is ”attracted” by xyz via the other
generator y 2 − z of J. Summing up the residual monomial and all the results of
the backward reduction we then obtain g 0 = x3 y + xy 3 + xyz whose dual D(g 0 ) =
1
∂x3 ∂y
6
+ 16 ∂x∂y 3 + ∂x∂y∂z is actually the noetherian operator of J relative to xyz.
The choice of the residual monomial xyz in Example 7.12 is not random. Indeed it
is maximal among all the residual monomials with respect to the derivative morphisms
(7.10).
Definition 7.22. Let m be a residual monomial of R/I. We say that m is a corner
monomial if it is maximal with respect to the monoid structure of Tn , i.e. if
xi · m ∈ LT(I), for all i = 1 . . . n.
If we represent R/I as a subset of Nn , the corner monomials are exactly in corner
position. Proposition 7.6 says that the noetherian operators are generated by the
ones corresponding to the corner monomials by taking the closure with respect to the
morphism 7.5. This fact allows to come up with a general procedure that constructs
the noetherian operators starting with the corner monomials and then generates the
entire space of noetherian operators.
Algorithm 7.6. Let I ⊂ R be a zerodimensional primary ideal of multiplicity µ
centered at the origin. The following list of instructions construct the noetherian
operators associated to I:
Input: a Gröbner Basis G of I and the residual monomials of R/I.
215
Output: the space of noetherian operators associated to I.
- Construct the set C of corner monomials using definition 7.22.
- For each corner monomial m ∈ C find the associated noetherian operators by rewriting it backwards with respect to G using definition7.21. Stop
when the backward reduction is not possible anymore or when the degree
of the polynomial obtained is µ − 1.
- Collect all the polynomials obtained in the set D.
- Compute the closure of D by applying the morphism σxi , i = 1 . . . n to
all its elements.
- For each element L in the closure of D calculate D(L).
The procedure 7.6 has been implemented on CoCoA using two main procedures.
The first performs the backwards reduction of a monomial using a set of generators
for I. The second finds the σ-closure of a set of operators. In the next section we
will analyze the performance of such algorithms compared to the ones discussed in
the previous sections. We also refer the reader to our web page [28] for an updated
list of examples and CPU times of the experiments run on our machines.
7.8
Some remarks on complexity
We now discuss the computational complexity of the algorithms of this chapter. Consider an ideal I of R to which we apply one of the algorithms of the previous sections,
and let G be a Gröbner Basis for I. The number of operations will be counted with
respect to the following parameters:
216
n = the number of unknowns of the polynomial ring R,
t = the cardinality of the Gröbner Basis G of I,
M = the maximum length of an element gi ∈ G,
µ = the multiplicity of I.
Complexity of Algorithm 7.2. After defining L0 = 1, the first step consists in
finding a linear operator L1 . This is done solving a system of t equations in the n
variables given by the coefficients cj of the representation
L1 =
n
X
cj ∂xj .
j=1
This is well known to have a complexity of order n3 . It is sufficient to find just one
solution to the system. The remaining µ − 2 steps proceed in a similar manner. Let
us consider the i-th step. We have to choose a multi-index α in {1 . . . n}i−1 and solve
the system of equations given by the conditions
Li (gk )|(0,...,0) = 0,
k = 1...t
(7.18)
where the i-th operator is given by the recursive formula
Li =
i−1
X
cj %xαj Lj
j=0
and the cj are unknown. Again, we have t equations in the i variables cj . This has a
complexity of O(i3 ). Consider how the closure condition 7.18 is written. For each k
we need at worst M · λi operations for the calculation of Li (gj ), where λi is the length
of Li . Besides this, we need to evaluate at the origin, which amount to n operations.
217
We then need an upper bound for λi . Since the length of L0 is 1 and the length of L1
is at most n, and at each step the new operator Li has length at most the sum of the
lengths of the previous ones, the sequence λi is bounded, pointwise, by the sequence
(1, n, 2n + 2, 4n + 4, . . . , 2i−1 (n + 1), . . . ).
Then λi ≤ 2i−1 . This means that the i-th step consists of O(ni−1 ·t·M ·2i−1 ) operations,
each requiring a solution of system of equation whose complexity is O(i3 ) = O(n3 ).
The total is then given by
O(n3+i · t · M · 2i−1 = O(n3 (2n)i tM ).
The above upper bound is valid for i = 1 as well as one can check directly. We
conclude that the total complexity for Algorithm 7.2 is given (asymptotically) by
C7.2
µ−1
X
1 − (2n)µ
= O(
tM n3 (2n)i−1 ) = O(tM n3
) = O(tM nµ+2 ).
1
−
2n
i=0
(7.19)
We notice that the complexity of this algorithm is exponential in the multiplicity of
the ideal. This makes this routine particularly slow when considering ideal of high
multiplicity, even with a small number of variables n.
Complexity of Algorithm 7.3. The algorithm presented in section 7.4.2 consists
of 3 basic steps:
1 - writing a Taylor polynomial T of degree µ − 1 with generic coefficients,
2 - reducing such polynomials using the Gröbner Basis of I,
3 - substituting each coefficient of the remainder with an operator using the fact that
218
they are Taylor coefficients.
The Taylor polynomial constructed at step 1 has a total of
µ
¶ µ
¶
µ
¶
n+1
n+2
n+µ−1
sµ (n) = 1 + n +
+
+ ··· +
2
3
µ
monomials, where each term is the number is monomials of a given degree. The
support of the polynomial has then cardinality sµ (n), which is a polynomial in n
of degree µ. Therefore we may consider the complexity of step 1 to be O(nµ . The
second step os just a series of t one-step reduction of each term in T . Each reduction
step has weight at most M , so we can conclude that we need to perform O(tM )
operations for each term of T . This makes a total of O(tM nµ ) operations for step
2. The last step is the transformation of each of each coefficient in the normal form
of T into an operator. There are µ coefficients and the table of conversion between
coefficients and operators stores O(nµ ) pairs (coefficient,operator). This amount to a
total complexity for algorithm 7.3 of
C7.3 = O(nµ + tM nµ + tnµ ) = O(tM nµ ).
(7.20)
Complexity of Algorithm 7.4. The algorithm for the computation of Noetherian
operators in the case of zerodimensional modules has exactly the same structure of
Algorithm 7.3, we the only difference that one has to take into account the fact that T
is a vector and all the generators of I are written as vectors, i.e. the also contains the
module components ei . For a submodule M of Rs , the complexity of this algorithm
will then be
C7.4 = O(stM nµ )
219
which has again the exponential nature encountered so far.
Complexity of Algorithm 7.5. The case of a positive dimensional ideal can be
treated as follows. The complexity is strictly related to C7.3 as Algorithm 7.5 is just
a variation of 7.3. The main difference is that the reduction step 2 may have to
be performed several times before we reach a final expression of the normal form
of T . In the worst case, we can assume that the degree of T decreases by one at
each reduction step, so the repeat loop has at most µ iterations. This is actually
confirmed experimentally by the tests we ran. Therefore, the complexity for the
positive dimensional case is
C7.5 = O(µtM nµ ).
Complexity of Algorithm 7.6. The idea of the last algorithm presented in the
chapter is to optimize Algorithm 7.3 by calculating the backwards reduction only for
corner monomials. Let c < µ be the total number of corner monomials as described in
the previous section. Each corner monomial requires a backward reduction up to, at
most, the degree µ − 1. We can expect this reduction to be as heavy as a reduction in
the usual way, since the number of steps required to ”climb up” the Taylor polynomial
are as many as the steps required to reduce the higher terms down to the residual
monomials. However, since this happens only for c terms, we can set the complexity
of the backwards reduction to be O(cM nµ ). The following step is the generation of
the entire space δ(I) using the closure condition. This has O(nµ ) operations. Then
the overall complexity for Algorithm 7.21 is
C7.21 = O(cM nµ + nµ ) = O(cM nµ ).
220
Remark 7.14. The complexities C7.2 , C7.3 , C7.4 and C7.21 calculated in this section
are all exponential in the multiplicity of the ideal (or module). Let us check, at least
for algorithm 7.3, that those are values that can be actually achieved. Consider the
ideal I = (x2 − y, y 2 ). Here t = 2, M = 2, n = 2 and µ = 4. The Taylor polynomial
has 10 monomials. The reduction steps performed are 6 as discussed in Example 7.6
and the final transformation of the coefficients into operators consists of browsing a
table of 10 elements 4 times. The total number of steps is then 56, which has the
same order of C7.3 = tM nµ = 64. If we try to construct examples of ideals where
the number of generators t and their maximum length M are ”big”, the multiplicity
of the ideal becomes too small to be significative. This is the case, for instance, of
the ideal J = (x1 , x1 + x2 , . . . , x1 + x2 + · · · + xn ) which is nothing but the maximal
ideal (x1 , . . . , xn ). Even if n = M = t and these can then be chosen arbitrarily big,
the multiplicity is just 1. A possible variation would be to increase the degree of the
generators, to make the multiplicity higher, but this in general increases the speed
of the reduction step, balancing for the higher multiplicity. In conclusion, I guess
that though the multiplicity has an exponential nature, in many actual cases the
performance of Algorithm 7.3 is much better than that. Its optimization 7.21 then is
the best choice for the case of a zerodimensional ideal as our results from [28] show.
Remark 7.15. Algorithm 7.2 does not present this advantage of being practically
less expensive than the complexity C7.2 shows. In fact, the exponential complexity
is almost always achieved. The reason is that the choice of the multi-index is done
without any particular rule. Simply, the set {1 . . . n}i−1 is browsed according to the
way the machine stores it, and there is only one choice of a multi-index that makes the
closure conditions solvable with a non-trivial solution. One cannot assume, then, that
221
the correct α is found right away. This part being the one that causes the complexity
to be exponential, it is clear that the factor nµ is not merely an upper bound for the
”worst case scenario”, but rather an ”average case” estimate of the complexity.
Chapter 8: Future research and open problems
This final chapter is devoted to the discussion of some open problems of research
that arise from the study of linear constant coefficient partial differential operators.
Some of them have been suggested in [25] or are problems we came across during our
recent experiments but we were not able to address because of time constraints and
capabilities of the machines we used, while some were inspired by some conversations
we had with other mathematicians. The first section collects those problems of a more
analytic nature. The second section lists some more exquisitely algebraic questions.
8.1
Analytical problems
In the setting of Dirac operators as discussed in Section 2.4, and especially with the
calculation of megaforms, we have pointed out many times how the radial relations
[{∂x i , ∂x j }, ∂x k ] = 0
play an important role in the construction of the syzygies of the Dirac system. The key
ingredient that allows to conclude that all the syzygies come from the radial relations
above (such as required in Chapter 6) is a particular polynomial decomposition called
Fischer decomposition. Let us briefly summarize it here. Given a Clifford polynomial
Q : (Rn )k −→ Cn , it can be shown (see [25]) that it admits a decomposition of the
type
Q(x1 , . . . , xk ) = M0 (x1 , . . . , xk ) + · · · + Mr (x1 , . . . , xk )
222
(8.1)
223
where M0 is monogenic in each Clifford variable and each polynomial Mj , j > 0, may
be written as a linear combination of products of the form
xs1 · · · xsj P (x1 , . . . , xk ),
∂xi P (x1 , . . . , xk ) = 0,
i = 1 . . . k.
Equation (8.1) is called rough Fischer decomposition. Note that the terms Mj of
degree j > 1 are not unique in general, while M0 and M1 are. In particular, M1 =
x1 P1 + · · · + xk Pk . If we denote by hxi , xj i the usual inner product in Rn , we have
the following
Conjecture 8.1. Every Clifford polynomial Q in k variables in a Clifford algebra Cn ,
with m ≥ 2k − 1 admits a unique decomposition of the form
Q(x1 , . . . , xk ) =
X
xs11 · · · xskk
si ,sij
Y
hxi , xj isij Mij (x1 , . . . , xk )
i<j
where the sum runs over all ordered tuples si , sij such that i = 1 . . . k, j = i + 1 . . . k
and the functions Mij are monogenic.
In the case of three Dirac operators in Cn , n > 5, the validity of Conjecture
8.1 allows to describe the syzygies via the decomposition of the polynomials in the
equations
x1 P1 (x2 , x3 ) + x2 P2 (x1 , x3 ) + x3 P3 (x1 , x2 ) = 0
and
x1 x2 P12 (x3 )+x2 x1 P21 (x3 )+x1 x3 P13 (x2 )+x3 x1 P31 (x2 )+x2 x3 P23 (x1 )+x3 x2 P32 (x1 ) = 0
which are the dual of compatibility conditions for the Dirac system. Note that we
are assuming also that such syzygies are at most quadratic. Proposition 4.3.6 in [25]
224
proves that the Fischer decomposition of Pi and Pij is sufficient to conclude that all
the syzygies are radial. We then propose the following
Problem 8.1. Prove that the Fischer decomposition holds in the general form of
Conjecture 8.1 and find the values of k and n for which the system of k Dirac operators
in Cn has only radial syzygies.
Besides being able to describe the nature of the syzygies using this approach, one
could take a more direct approach using Gröbner Basis and some of the techniques
used in [3]. The aim is to describe the algebraic properties of the module associated
to k Dirac operators for the algebraic analysis of the Dirac system.
Problem 8.2. Let k, n be positive integers and let M be the module associated to the
Dirac operator in k Clifford variables in Cn . Using a Gröbner Basis for the module
Mn,k , calculate an explicit expression for the generators of LT(Mn,k ) and derive the
Hilbert series HMn,k (t), the length of the complex, the (graded) Betti numbers, the
cohomologies and the Castelnuovo-Mumford regularity.
The compatibility conditions for the Dirac system with 3 operators are described
in Theorem 2.9. Since they involve only three Dirac operators at a time, the relations
∂xj ∂xi gi − ∂xj i gj = 0,
(8.2)
{∂xi , ∂xj }gk − ∂xk ∂xi gj − ∂xk ∂xj gi = 0.
(8.3)
actually describe the first compatibility conditions for any number of Dirac operators,
assigning to i, j and k all the possible distinct values. This is experimentally confirmed
by the calculation of the syzygies in the case of 4 operators both using the real
representation as in Section 2.4 and through megaforms. However, it is evident from
225
the subsequent syzygies, already in the case of 3 operators, that the higher order
compatibility conditions involve more than just 3 of the relations obtained at the
previous step, so the description of Theorem 2.9 is not applicable to the general case.
Problem 8.3. Find an explicit expression for the (radial) syzygies for the complex
associated to k Dirac operators, for k > 3.
The use of CoCoA for the computation of the syzygies of the Cauchy-Fueter system
exploits the real representation of the operator and hence produces syzygies that are
hard to reconstruct in terms of quaternionic-type relations. The approach described
in Chapter 5 cuts the dimension by a factor of two and allows to explicitly represent
all the compatibility conditions on terms of complex operators. We know for a fact
that at each step some of the relations are not expressible in terms of quaternionic
operators, as some new symbols like D0 , D00 and Ď have to be introduced for the
exceptional syzygies. The following then still needs to be addressed
Problem 8.4. Find a quaternionic representation of the compatibility conditions of
the Cauchy-Fueter system in n operators utilizing a maximal number of Cauchy-Fueter
operators, and their conjugate, and identify all the exceptional syzygies.
In view of Theorem 2.5 there is a class of operators for which the Hilbert resolution
is dual to itself in the sense given in Definition 2.1. This fact allows a description of
the dual of the space of hyperfunctions in terms of the original system. The CauchyRiemann operator in n variables and the Dirac operator in 3 variables, when the
dimension of the Clifford algebra is high enough to allow only radial syzygies, are
examples of self-dual operators. For explicit examples in which the dimension of the
Clifford Algebra and the number of operators are fixed it is easy, at least in theory,
226
to verify if an operator is self–dual and then derive the description of the space of
hyperfunctions. It suffices to calculate the free resolution with CoCoA and verify that
the cohomologies vanish except at the last step. As Table 2.2 shows, however, we
have not been able to describe the Dirac complex for 4 operators in the radial case,
so we still do not know if the assumption that this complex is self-dual is feasible.
The general conjecture about the radial Dirac complex is that its length is 2k + 1 and
that the last map is the transpose of the original (note that the conjugate of the Dirac
operator is its opposite because the matrix symbol is antisymmetric). We hope to be
able in the future to perform some more computational experiments for the case of 4
and 5 operators, so we propose the following
Problem 8.5. Consider the Dirac system associated to k operators in real dimension
n. Provided that for n > 2k − 1 the associated complex contains only radial syzygies,
prove that the length of the complex is s = 2k + 1
1
and that the Dirac operator is
self-dual. Conclude that the space of monogenic functions in k Clifford variables is
strongly dual to itself in this case.
As widely discussed in [25], Theorem 4.1 and Corollary 4.1 have a significant
number of analytical consequences. In particular they allow an immediate proof of
the Hartogs’ phenomenon for biregular functions in 2n quaternionic variables (the
case n = 1 had already been proved in [13]). They also allow the construction of
a hyperfunction like theory for boundary values of such functions. In general, like
for the case of regular functions of n quaternionic variables, we do not expect the
complex to be self-dual, as the last map usually involves more operators than the
1
Updated January 2006: new computations done by P. Franek suggest the the length of the Dirac
complex is actually quadratic in the number of operators.
227
generators of the system. Nonetheless, Theorem 4.1 provides the vanishing of all the
Ext modules except the last one. This leads to think that an analogue of Theorem
5.6 and its consequences can be stated for biregular functions:
Problem 8.6. Consider the system 4.7 whose sheaf of solutions is the space of biregular functions in 2n quaternionic variables as introduce in Chapter 4. Describe explicitly the last map in the free resolution of the operator and deduce a duality theorem
for the space of hyperfunctions.
In its fundamental work [35] Ehrenpreis developed a theory of LAU spaces which
roughly speaking are spaces X of generalized functions for which there exists an isomorphism between X 0 and a space of entire functions with suitable growth conditions.
The interest of these spaces lies in the fact that constant coefficient differential operators are surjective among these spaces and solutions of homogeneous systems in
these spaces can be represented by convolution integrals on the characteristic variety. A consequence of the definition is that kernels of constant coefficient differential
operators in LAU spaces are still LAU. For this reason, if we think of the space of
regular functions on H as the space of functions in (C ∞ (R4 ))4 satisfying the CauchyFueter system, we know that it is LAU. In Chapter 3 we proved directly that suitable
quaternionic operators are surjective on such a space (see Corollary 3.4 and following
results). Let p(q) be a polynomial in one quaternionic variable and denote by ∂/∂q
the conjugate of the Cauchy-Fueter operator. The previous considerations lead us to
formulate the following:
Problem 8.7. See whether it is possible to prove an integral representation theorem
in the spirit of Ehrenpreis for regular functions f such that p(∂/∂q)f = 0 without
228
using the real representation of the operators, i.e. with the only use of quaternionic
noetherian operators of the type f (∂/∂ q̄, ∂/∂q) and with characteristic variety
V (q) = {q ∈ H | p(q) = 0}.
8.2
Algebraic problems
Let R = C[x1 , . . . , xn ] and consider the module M = Rs /hP t i associated to a system
of linear constant coefficient partial differential equations P (D)f = 0. The algebraic
analysis of the system passes through the study of the algebraic properties of M . In
particular all the algebraic objects we look for are somehow related to the Hilbert
series of M . A result of Macaulay (see [54]) states that the quotient C-space Rs /hP t i
is isomorphic (where the isomorphism has degree zero) to the space Rs / LThP t i,
which is in general much easier to study and characterize than the module M itself.
It is an immediate consequence of the fact that LThP t i is generated by monomials
in Tn he1 , . . . , es i that it is of the form I1 e1 ⊕ · · · ⊕ Is es , where Ii is an ideal of R
and ei is the canonical generator of Rs , i = 1 . . . s. Then the Hilbert series of M , by
additivity, depends on the Hilbert series of each ideal Ii . In the case of the CauchyFueter operator (see [3]) and as proved in Chapter 4 for biregular functions in several
quaternionic variables, the form of the monomial module LThP t i is diagonal (see
Section 4.3), i.e. Ii = I for all i = 1 . . . s. Since computations with CoCoA show that
this is the case for the Dirac operator as well, we propose to
Problem 8.8. Prove that the leading term module of the module associated to the
Dirac operator is diagonal and use this fact to calculate the Hilbert function of the
module.
229
This will allow to also complete the study of the Dirac operator from an algebraic
point of view as in Problem 8.2. A more general formulation of Problem 8.8 can also
be stated as follows:
Problem 8.9. Classify the linear constant coefficients operators P (D) such that the
leading term module LThP t i is diagonal.
It seems reasonable to conjecture that the algebraic properties of the module M
associated to P (D) also depend on the nature of the blocks of P , as discussed in
Chapter 3. The results discussed in this chapter are require such blocks to commute,
which may be considered as a restrictive condition on P , while their regularity seems
to be a fair assumption for us since each block involve a different set of variables.
It is worth to mention what can be done in the non commutative situation. What
follows was developed by Kawai and Takei [49] to deal with the case of several non
commuting matrices. Let us start with two matrices Q1 , Q2 and let us introduce the
commutators
R0 = [Q1 , Q2 ],
R1 = [Q1 , R0 ],
R2 = [Q2 , R0 ].
It can be directly proved that the system associated to (Q1 , Q2 , R0 , R1 , R2 ) is equivalent to the system associated to (Q1 , Q2 ). Let us introduce the notation
Lj = Qj ,
for j = 1, 2;
Lj = Rj−3
for j > 2.
We will say that the pair (Q1 , Q2 ) is weakly commutative if there exist polynomials
cjkl such that for all j, k
[Lj , Lk ] =
X
1≤l≤5
cjkl Ll .
(8.4)
230
This notion of weak commutativity allows to construct the complex: for example, the
first step of the resolution is a 10 × 5 matrix whose entries are matrices computed
using the relation (8.4). Note that this approach is extremely useful, see [49], though
it does not allow to treat systems like the Cauchy–Fueter system or the Dirac system
in several variables; in fact in those cases, the procedure does not recover not even
the first syzygies of the system. Additional ideas concerning this approach can also
be found in [53].
Problem 8.10. Consider an operator satisfying the weakly commutativity property
and use relations (8.4) to construct a resolution of the associated module. Compare
the results obtained with this method with the Hilbert free resolution.
An alternative way to construct a resolution for a polynomial map defined by a block
matrix, when the blocks do not commute, is to use a general tool that produces a
complex using maps defined only in terms of the entries aij of the matrix P . The
result of this construction is called a generalized Koszul complex. There are slightly
different versions of such construction. The most common two are due to J. Eagon
and D.G. Northcott [31] and D.A. Buchsbaum [17]. Although those complex are fairly
easy to describe in terms of tensor products and linear operations on the rows and
the columns of the matrix P , quite often they do not provide a minimal resolution,
and the cohomologies are not necessarily trivial. We propose to
Problem 8.11. Construct a generalized Koszul complex for the modules associated
to the Cauchy-Fueter operator and the Dirac operator, and compare the results obtained with the Minimal Hilbert Free resolution. Establish if some of the properties of
the regular functions of several quaternionic variables and monogenic functions on a
231
Clifford algebra can be deduced form such complexes.
The algorithmic procedure to construct the Dirac complex using megaforms still
presents some open problems to be addressed as discussed in the final section of
Chapter 6. A first important step could be to
Problem 8.12. Optimize Algorithm 6.1 by providing an automatic way to choose a
minimal set of syzygies at each syzygy step. Consider also a more efficient way of
performing radial reductions.
The next problem is of a more theoretical nature and is inspired by the description
of the Dirac operators as generators of the radial algebra:
Problem 8.13. Generalize the construction of the space of megaforms to the case
of k operators generating an associative algebra and satisfying a finite set of polynomial relations. Develop a full theory of non commutative syzygies following this
construction and implement their explicit calculation on a computer algebra software.
Let us conclude this list of research problems with two questions on Noetherian
operators. From a conversation with B. Sturmfels and given the results presented the
last chapter of his work [69] in which he proposes a method to construct explicitly
the noetherian operators for any monomial ideal, without the assumption of finiteness
of the set of residual monomials and without any use of primary decomposition, it
becomes relevant to study possible links between the set of operators associated to an
ideal an the operators constructed for the monomial ideal generated by the leading
terms. However, some considerations like the one discussed in section 7.3 make us
believe that there is no way to avoid primary decomposition, but the optimization
232
presented in 7.7 shows clearly that in the zerodimensional case there is a tight relation
between the noetherian operators and those of the leading term ideal. Thus the
problem we want to consider is
Problem 8.14. Let I be a primary ideal of R and consider the noetherian operators
associated to LT(I) as constructed in [69]. Find an algorithm that allows to compute
the noetherian operators of I given the ones of LT(I). Specify in which cases this can
be done finding suitable conditions on I.
233
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234
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Curriculum Vitae
Alberto Damiano was born on December 4, 1978, in Sanremo, Italy, and is an Italian citizen. He graduated from Liceo Scientifico G. Saccheri (scientific high school),
Sanremo, Italy, in 1997. He received his Laurea (Master) in Mathematics from the
University of Genova in 2001. He as been a graduate research assistant for the Dean
of the College of Arts and Sciences at George Mason University, Fairfax, VA (United
States) and a teaching assistant for the Department of Mathematics. During his
permanence in the United States, he participated in many scientific conferences and
seminars both with poster presentations and as a speaker. In May 2004 he organized
and participated in the George Mason conference in honor of Carlos Berenstein’s
mathematics sponsored by the College of Arts and Sciences. He has been awarded for
three years in a row, 2003, 2004 and 2005, with the GMU Vision Award for academic
excellence and in 2003 he has been recognized with the Who’s Who among graduate
students. His scientific publications include:
- J. Bureš, A. Damiano, I. Sabadini, Explicit resolutions for the complex of several
Fueter operators, submitted to J. Geom. Phys. 2005.
- F. Colombo, A. Damiano, Inverse problems for heat source equations, to appear on
Houston Journal of Math, 2006.
- F. Colombo, A. Damiano, I. Sabadini, D.C. Struppa, A surjectivity theorem for differential operators on spaces of regular functions, Complex Variables, (50) 2005 no.
6, 389-400.
- A. Damiano, I. Sabadini, D.C. Struppa, Computational methods for the construction
of a class of noetherian operators, submitted 2004.
- A. Damiano, I. Sabadini, D.C. Struppa New algebraic properties of biregular functions in 2n quaternionic variables, submitted 2004.
240
- F. Colombo, A. Damiano, I. Sabadini, D.C. Struppa, Quaternionic hyperfunctions
on 5-dimensional varieties in H2 , preprint 2005.
- F. Colombo, A. Damiano, I. Sabadini, D.C. Struppa, A new Dolbeault complex in
quaternionic and Clifford analysis , preprint 2005.
- L. Kerschberg, M. Chowdhury, A. Damiano, H. Jeong, S. Mitchell, J. Si, and S.
Smith, Knowledge Sifter: Agent-Based Ontology-Driven Search over Heterogeneous
Databases using Semantic Web Services, Proceedings of International Conference on
Semantics for a Networked World, Semantics for Grid Databases, Maison des Polytechniciens, Paris France, 2004, pp. 276-293.
- S. A. Mitchell, J. Durgavich, A. Damiano, S. J. Smith, R. MacCracken Use of Genetic Algorithms With Multiple Metrics Aimed At the Optimization of Automotive
Suspension Systems, presented at Motorsports Engineering Conference and Exposition, November 2004, Dearborn, MI, USA.
Besides mathematics, his interests include playing the piano, traveling and being
an active ally to the Gay and Lesbian community.