SPECTRALLY ARBITRARY ZERO-NONZERO PATTERNS OF MATRICES OVER
A VARIETY OF FIELDS
By
Timothy C. Melvin
A dissertation submitted in partial fulfillment of the
requirements for the degree of
DOCTOR OF PHILOSOPHY
WASHINGTON STATE UNIVERSITY
Department of Mathematics
December 2013
ii
To the Faculty of Washington State University:
The members of the Committee appointed to examine
the dissertation of Timothy Melvin find it satisfactory
and recommend that it be accepted.
Judith McDonald, Ph.D., Chair
Michael Tsatsomeros, Ph.D.
Matthew Hudelson, Ph.D.
iii
SPECTRALLY ARBITRARY ZERO-NONZERO PATTERNS OF MATRICES OVER
A VARIETY OF FIELDS
Abstract
by Timothy C. Melvin, Ph.D.
Washington State University
December 2013
Chair: Judith J. McDonald
An n × n zero-nonzero pattern A is spectrally arbitrary over a field F if for each
monic polynomial p(t) = tn + a0 tn−1 + · · · + an ∈ F[t], there exists a matrix A over F
with zero-nonzero pattern A such that pA (t), the characteristic polynomial of A, equals
p(t). The nilpotent-Jacobian method is the most widely used method to determine if a
zero-nonzero pattern is spectrally arbitrary over R or C, but this method only works over
topologically complete fields. Analyzing the proof of the nilpotent-Jacobian method, we
explore methods for determining if a zero-nonzero pattern is spectrally arbitrary over
field extensions of Q. Using these methods, we classify all 2 × 2 patterns, 3 × 3 patterns,
and all 4 × 4 complex minimally spectrally arbitrary patterns over extensions of Q.
We prove that if a system of multivariate polynomials with k variables with coefficients from an algebraically closed field E has a solution in Lk , where L ⊇ E, then there
will be a solution in Ek . Furthermore, if there is a strictly nonzero solution in Lk , then
there is a strictly nonzero solution in Ek . Thus, any n × n complex spectrally arbitrary
pattern will be spectrally arbitrary over the algebraic closure of Q.
A pattern B is a subpattern of A if aij = bij whenever bij 6= 0, and B is a superpattern of A if aij = bij whenever aij 6= 0. If for each monic polynomial p(t) ∈ F, there is a
iv
matrix B over F whose zero-nonzero pattern is a subpattern of A such that pB (t) = p(t),
we say A is relaxed spectrally arbitrary over F. We prove that every pattern that is
spectrally arbitrary over a dense subfield of R or C is relaxed spectrally arbitrary over
R or C, respectively. We use this result to show that the minimum number of nonzero
entries of a spectrally arbitrary pattern over any field extension of Q is 2n − 1.
It has been conjectured that if a pattern is spectrally arbitrary over F, then each
of its superpatterns is spectrally arbitrary over F. While we explore a number of counterexamples to this superpattern conjecture over the finite field F3 , it remains an open
question over all other fields.
v
Acknowledgment
I first and foremost want to thank my wife Jessica for supporting me, loving me, and
putting up with me. Without you, I do not see how this dissertation would be possible.
This is as much your effort as mine.
I would like to thank my advisor, Judith McDonald, for her support and guidance
the past four years. I attempted to research just about anything but linear algebra, but
Judi’s patience, professionalism, selflessness, and passion for linear algebra continually
drew me back to the study of spectrally arbitrary matrix patterns. My committee
members, Michael Tsatsomeros, and Matt Hudelson, provided valuable insight, and I
had many meaningful conversations with my fellow graduate students at Washington
State including Elizabeth Bodine, Amy Streifel, Sharif Ibrahim, Jimmy Burk, and Eric
Larson. I want to give a special thanks to Nicholas Crabb as the hours spent discussing
spectrally arbitrary patterns, mathematics, and life in general while drinking beer with
you was invaluable to this work. Good luck with your PhD at UC Davis.
I first cultivated a love for mathematics whilst pursuing my Master’s degree at Sacramento State University. I want to thank some of my teachers and fellow graduate students at Sacramento State for pushing me academically and stimulating me intellectually including Tracy Hamilton, Bin Lu, Gary Shannon, Kecheng Zhou, Charles Albright,
Sang Sertich, Phuong Le, Toni Cravens, and Dane Fleshmen.
Finally, this paper could not have been done without the love and support of my
family. My mom encouraged me to pursue my dreams, whatever they may be. My
dad always encouraged me to think outside the box, and my step-dad instilled in me a
respect for hard work. My sister called me “nerd” a lot growing up (and still does), but
I think that actually encouraged me to pursue mathematics.
vi
Contents
1 Introduction, Definitions and Notation, and Motivational Questions
1
1.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.2
Definitions and Notation . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
1.2.1
Matrix Pattern Definitions . . . . . . . . . . . . . . . . . . . . . .
3
1.2.2
Field Theory and Polynomial Rings . . . . . . . . . . . . . . . . .
5
1.2.3
The Digraph of a Matrix . . . . . . . . . . . . . . . . . . . . . . .
9
Motivational Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
1.3
2 The Jacobian Method and the Surjective Neighborhoods Condition
17
2.0.1
The Proof of the nilpotent-Jacobian Method . . . . . . . . . . . .
18
2.0.2
Surjective Neighborhoods Condition . . . . . . . . . . . . . . . . .
20
2.0.3
The Nilpotent-Jacobian Method over Extensions of Q . . . . . . .
22
3 Small Patterns
28
3.1
2 × 2 Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
3.2
3 × 3 Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
4 4 × 4 Patterns
4.1
33
Real Spectrally Arbitrary Patterns . . . . . . . . . . . . . . . . . . . . .
34
4.1.1
Real Spectrally Arbitrary Patterns with 8 Nonzero Entries . . . .
34
4.1.2
Real Spectrally Arbitrary Patterns with 9 Nonzero Entries . . . .
39
vii
4.2
Minimally Spectrally Arbitrary Patterns over C That Are Not Spectrally
Arbitrary Over R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
4.2.1
M4 and the Surjective Neighborhoods Condition . . . . . . . . . .
42
4.2.2
M4 Over Q and Extensions of Q . . . . . . . . . . . . . . . . . .
45
4.2.3
Fill 1 Superpatterns of M4 . . . . . . . . . . . . . . . . . . . . . .
48
4.2.4
Fill 2 Superpatterns of M4 . . . . . . . . . . . . . . . . . . . . . .
52
5 Field Extensions, The 2n Conjecture, The Superpattern Conjecture
60
5.1
Spectrally Arbitrary Patterns and Field Extensions . . . . . . . . . . . .
60
5.2
The Superpattern Conjecture . . . . . . . . . . . . . . . . . . . . . . . .
64
5.3
5.2.1
4 × 4 Counterexamples to the Superpattern Conjecture over F3
5.2.2
5 × 5 and 6 × 6 Counterexamples to the Superpattern Conjecture
66
Lower Bound of Nonzero Entries in Spectrally Arbitrary Patterns . . . .
69
5.3.1
The 2n Conjecture and Finite Fields . . . . . . . . . . . . . . . .
69
5.3.2
Lower Bound of Nonzero Entries in Spectrally Arbitrary Patterns
.
over Extensions of Q . . . . . . . . . . . . . . . . . . . . . . . . .
64
71
6 Concluding Remarks
74
A 4 × 4 Minimally Spectrally Arbitrary Patterns Over R or C
76
A.1 4 × 4 Real Minimally Spectrally Patterns with 8 Nonzero Entries . . . .
76
A.2 4 × 4 Real Spectrally Patterns with 9 Nonzero Entries . . . . . . . . . . .
85
A.3 The Pattern B1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
B Sample Mathematica Code
104
viii
C Sample Sage Code
106
Bibliography
109
1
Chapter 1
Introduction, Definitions and Notation, and
Motivational Questions
1.1
Introduction
Combinatorial matrix theory is the study of properties of matrices when only combinatorial or algebraic information is known, as opposed to numerical or analytical information.
Oftentimes, data may be incomplete or exact numerical values will be unknown, but it
will be known whether a value is 0, positive, or negative such as in a network flow. It
is still important to be able to analyze such systems with mathematical rigor. Combinatorial matrix theory strives to analyze such “incomplete” matrix patterns to answer
questions about the properties of the matrix, such as rank, singularity, inertia, eigenvectors, stability, inverses, and spectra by drawing on a wide range of disciplines such
as combinatorics, graph theory, field theory, algebraic geometry, commutative algebra,
differentiable geometry, topology, and linear algebra. It is also important to be able
to analyze systems that require a degree of roundoff error that occur in many computational and numerical methods. Combinatorial matrix theory provides inscrutable
mathematical certainty to systems with uncertain data. This has been an incredibly
fertile area of research, adding to the body of knowledge of linear algebra along with
many others branches of mathematics.
2
A zero-nonzero (respectively, sign) pattern A is a square matrix whose entries are
elements of the set {0, ∗} ({0, +, −}), where ∗ denotes a nonzero entry. The study of
sign patterns first arose about 60 years ago as an application of economics, but it has
applications in other areas such as signal processing, ecology, chemistry, and geology.
Combinatorial matrix theory has been studied extensively in the past 60 years (see [21]),
and the study of sign patterns in particular has greatly contributed to the development
of qualitative theory of real matrices. Recently, analysis of zero-nonzero matrix patterns
over other fields such as C (see [3]), Q, and the finite fields (see [18], [11], [14]) was
initiated.
The definition of a spectrally arbitrary sign pattern was first introduced in a paper
by Drew, Johnson, Olesky, and van den Driessche[7] in 2000. An n × n zero-nonzero
(sign) pattern is said to be spectrally arbitrary over a field F if for each nth degree monic
polynomial r(t) in F[t], there exists a matrix A with zero-nonzero (sign) pattern A (A
is called a realization of A ) such that the characteristic polynomial of A is equal to
r(t). Since 2000 many techniques have been developed to determine whether a sign
pattern is spectrally arbitrary over R and C (see [24], [1], [10], [8], [3]). The first paper
on zero-nonzero patterns was published in 2005 by Corpuz and McDonald[6], where all
spectrally arbitrary 2 × 2, 3 × 3, and 4 × 4 patterns over R were classified. McDonald and
Yielding published a paper[3] in 2010 that classified all spectrally arbitrary 3 × 3 and
4×4 zero-nonzero patterns over C. Within the past few years, Bodine and McDonald(see
[15], [18]) worked on classifying spectrally arbitrary zero-nonzero patterns over various
finite fields and algebraic extensions of Q.
The nilpotent-Jacobian method, developed in [7], is the most commonly used method
3
to determine if a sign or zero-nonzero pattern is spectrally arbitrary over R or C. However, this method only works over a field that has a topologically complete method, as it
utilizes the contraction mapping principle. Thus, other methods are needed to determine
if a pattern is spectrally arbitrary over a finite field or an extension of Q. We develop one
sufficient method, the surjective neighborhoods condition, in chapter 2, to determine if
a pattern is spectrally arbitrary over any field with a defined metric, such as Q. We use
this method to classify all real and complex 3 × 3 and 4 × 4 spectrally arbitrary patterns over extensions of Q. By studying zero-nonzero patterns over extensions of Q we
analyze the relationship between the combinatorial structure of a pattern and the field
structure required in order for the pattern to be spectrally arbitrary over the field, and
in our analysis of small patterns we make observations that may lead to more general
results in the future. One objective of this work is to garner a better understanding of
the interplay between a matrix pattern and determining which particular field structure
or structures are necessarily in order for the pattern to be spectrally arbitrary. Thus,
given data from a particular field of numbers it may be possible to determine over which
field full spectra can be realized by a zero-nonzero matrix pattern.
1.2
1.2.1
Definitions and Notation
Matrix Pattern Definitions
Let F denote a fixed field and let S denote the set of symbols S = {0, ∗, ], +, −}, where ∗
denotes a nonzero field element, ] denotes an arbitrary field element, + denotes a positive
field element (if F is an ordered field), and − denotes a negative field element (if F is an
4
ordered field) (see [13]). For S ⊆ S, an S-pattern is an n × n matrix with entries from
S. In particularly, a {0, ∗}-pattern is called a zero-nonzero pattern, a {0, ]}-pattern is
called a relaxed zero-nonzero pattern, and a {0, +, −}-pattern is called a sign (or signed)
pattern. Since we will focus almost exclusively on zero-nonzero patterns throughout
this paper, all patterns will be understood to be a zero-nonzero patterns unless stated
otherwise.
A pattern B is a subpattern of a pattern A if aij = bij whenever bij 6= 0. A pattern
B is a superpattern of a pattern A if aij = bij whenever aij 6= 0. In particular, we say
a pattern B is a superpattern of A of fill j if it is a superpattern of A with exactly j
zero entries from A made nonzero.
For any S-pattern A = (αij ), a matrix A = (aij ) with entries from a field F is called
a realization of A over F if aij is nonzero whenever αij = ∗, and aij = 0 whenever
αij = 0. Note, if αij = ], then aij can be zero or nonzero. An n × n zero-nonzero
pattern is said to be a spectrally arbitrary pattern over F if for every monic polynomial
p(t) with coefficients from F of degree n, there exists a realization A of A such that the
characteristic polynomial of A, denoted pA (t), is p(t). Given a particular monic degree
n polynomial r(t) over F, we say an S-pattern A realizes the polynomial r(t) if there
exists a realization of A whose characteristic polynomial equals r(t).
0
Given a zero-nonzero pattern A , the relaxation of A is the matrix A 0 = (αij
) where
0
αij
= ] if and only if αij = ∗. We say A is a relaxed spectrally arbitrary pattern over F
if its relaxation is spectrally arbitrary over F. Clearly, every zero-nonzero pattern that
is spectrally arbitrary over F is also relaxed spectrally arbitrary over F.
A n×n zero-nonzero pattern A is said to be potentially nilpotent over F if there exists
a nilpotent realization A over F. Note that the relaxation of any pattern is potentially
5
nilpotent, because we can set each ] entry in A equal to 0. By definition each spectrally
arbitrary pattern is potentially nilpotent, but the converse is not necessarily true (see
M4 in Chapter 4).
A pattern A is said to be minimally spectrally arbitrary over F if it is spectrally
arbitrary over F, but none of its subpatterns are spectrally arbitrary over F. A pattern
A is said to be combinatorially minimally spectrally arbitrary if A is spectrally arbitrary
over R or C and none of its subpatterns are spectrally arbitrary over any field.
Let A be an S-pattern with realization A = (aij ). For U = {ai1 j1 , . . . , aik jk }, a
set of nonzero entries of A, define the matrix C from A by replacing ail ,jl ∈ U with
indeterminate xl for each l = 1, . . . , k. Let pC (t) = tn + f1 tn−1 + · · · + fn be the
characteristic polynomial of C. We say F = (f1 , f2 , . . . , fn ) : Fk → Fn is the polynomial
map of A determined by U . If U includes each nonzero entry of A and aij is nonzero
whenever αij ∈ {∗, ]}, we say F is the polynomial map of A . The Jacobian matrix of F
(or Jacobian of F ), denoted by Jac(F ), is the n × k matrix whose (i, j) entry is
∂fi
.
∂xj
An n×n matrix is called a permutation matrix if there is exactly one 1 in each row and
column and 0 entries elsewhere. We say two S-patterns A and C are permutationally
similar if there is a permutation matrix B such that A = BC B −1 . Note if A and C
are permutationally similar, then A is spectrally arbitrary over a field F if and only if
C is spectrally arbitrary over F.
1.2.2
Field Theory and Polynomial Rings
Given a field F, we say F ⊆ E is an extension of fields (or E is an field extension of F)
if both F and E are fields under the same operations. Every extension E of F can be
6
considered as a vector space over F, so we define the dimension of E over F, denoted
dimF E, as the dimension of E as a vector space over E. For α1 , . . . , αn ∈ E, we denote
by F(α1 , . . . , αn ) the smallest (with respect to set inclusion) field extension of F that
contains α1 , . . . , αn . Thus, F ⊆ F(α1 , . . . , αn ) ⊆ E.
We say the characteristic of a field F is the smallest integer n such that n · a = 0 for
all a ∈ F. If no such integer exists, we say the field has characteristic 0. Note that all
finite fields have a prime characteristic, and all extensions of Q have characteristic 0.
The polynomial ring over F of k variables is denoted by F[x1 , . . . , xk ] (or F[X] if the
number of variables is understood). If for α ∈ E there exists a polynomial p(x) ∈ F[x]
such that p(α) = 0, then α is called algebraic over F. An extension E is called an
algebraic extension of F, if each element in E is algebraic over F. Every finite dimensional extension of fields is algebraic, but the converse is not true. Observe that
√ √ √ √
√
Q ⊂ Q 2, 3, 5, 7, . . . and Q ⊂ Q k 2 : k ∈ N are examples of infinite, algebraic field extensions. The algebraic closure of a field F, denoted by F, is the smallest
(with respect to set inclusion) field extension of F such that every nonconstant polynomial F ∈ F[x] has a root, e.g. R = C.
We recursively define a set of field extensions of Q as follows. Set Q0 = Q and
√
1
1
for all n ∈ N, we define Q+ n+1
= Q+ 1n
p : p ∈ Q+ 1n and p > 0 and Q n+1
=
2
2
2
2
√
1
Q 1n
p : p ∈ Q 1n . Thus, Q0 ⊂ Q+ 1 ⊂ Q+ 1 ⊂ · · · , and each Q+ n+1
is an infinite
2
2
2
4
2
field extension of Q+ 1n . Similarly Q0 ⊂ Q 1 ⊂ Q 1 ⊂ · · · , and each Q
2
2
4
1
2n+1
is an infinite
field extension of Q 1n . Both of these chains of field extensions are bounded above by Q,
2
so by Zorn’s Lemma, each has a maximal element. We denote the maximal element of
the first chain by Q√+ , and we denote the maximal element of the second chain by Q√ .
We recursively define another chain of field extensions of Q as follows: set Q0 =
7
√
Q, and for each n ∈ N set Q2n−1 = Q2n−2 p : p ∈ Q2n−2 , p > 0 and set Q2n =
√
Q2n−1 3 p : p ∈ Q2n−1 , p > 0 . Note that Q0 ⊂ Q1 ⊂ Q2 ⊂ · · · and this infinite chain
of fields is bounded above by Q. Thus, Zorn’s Lemma asserts that this chain has a
maximal element. We denote this maximal field by Qβ .
All numbers in Q√+ , Q√ , and Qβ are called constructible, so these fields are examples
of constructible fields (see [28]). Each of these fields are algebraic over Q as they are all
√
subfields of Q. By construction, if y ∈ Q√ , then y ∈ Q√ ; if y ∈ Q√+ and y ≥ 0, then
√
√
y ∈ Q√+ ; and if y ∈ Qβ and y ≥ 0, then y ∈ Qβ and 3 y ∈ Qβ .
An element β in a field extension of F that is not algebraic over F is called transcendental over F. For example, π is transcendental over Q, so π ∈
/ Q.
Let R be a ring and r1 , . . . , rk ∈ R. The ideal generated by r1 , . . . , rk is given by the
set
hr1 , . . . , rk i =
nX
ai ri : ai ∈ R and the sum is finite
o
A ring R is called Noetherian if each ideal in R is finitely generated. The Hilbert Basis
Theorem[19] states that if R is a Noetherian ring, then R[x1 , . . . , xk ] is a Noetherian ring.
Since the only ideals of any field are the field itself, generated by any nonzero element,
and the trivial ideal, {0} = h0i, F[X] is Noetherian by the Hilbert Basis Theorem.
√
Let I be an ideal in a ring R. We define the radical ideal of I, denoted by I, to be
√
the set a ∈ R : aN ∈ I for some N ∈ N . Clearly, I ⊆ I, but the inclusion may be
p
√
strict. For example hx2 i = hxi in Q[x]. We say an ideal J is radical if J = J.
Let F ⊆ E be an extension of fields. Given an ideal I = hg1 , . . . , gr i in F[x1 , . . . , xk ]
(or E[x1 , . . . , xk ]), we call VE (I) = p ∈ Ek : g(p) = 0 for all g ∈ I the affine algebraic
set generated by I. We usually write VE (g1 , . . . , gr ) instead of VE (hg1 , . . . , gr i). By
the definition of a generating set we have VE (I) = p ∈ Ek : gi (p) = 0 for i = 1, . . . , r .
8
Given a set of points U in Fk (or Ek ), we call the following set the ideal of U , IE (U ) =
{g ∈ E[x1 , . . . , xr ] : g(p) = 0 for all p ∈ U }. Note that it is common to drop the subscripts in VE (I) and IE (U ) when working over a common field. The following are wellknown results regarding algebraic sets, U and V , and ideals of a set of points (see [19]):
1. If K ⊆ J, then V (J) ⊆ V (K).
2. V (h0i) = Fk and V (h1i) = ∅.
3. For a1 , . . . , ak ∈ F, V (x1 − a1 , . . . , xk − ak ) = {(a1 , . . . , ak )}.
4. If U ⊆ V , then I(V ) ⊆ I(U ).
5. I(∅) = F(X).
6. If F is infinite, then I Fk = {0}.
7. For a1 , . . . , ak ∈ F, I ({(a1 , . . . , ak )}) = (x1 − a1 , . . . , xk − ak ).
8. K ⊆ I (V (K)) and the subset may be strict, e.g. K = hx2 + 1i ∈ R[x].
9. K ⊆
√
K ⊆ I (V (K)).
Hilbert’s Nullstellensatz is one of the central theorems in algebraic geometric. The
Nullstellensatz, which is German for “zero-locus-theorem”, establishes a fundamental
relation ship between the geometry of the zeros of a set of multivariate polynomials
and the algebra of the ideal generated by the set of polynomials in the corresponding
polynomial ring. Hilbert’s Nullstellensatz has two parts, a “weak” and “strong” version.
Theorem 1.1 (Hilbert’s Weak Nullstellensatz) Let K be an algebraically closed
field. If I is a proper ideal in the polynomial ring K[x1 , . . . , xk ], then V (I) 6= ∅.
9
Conversely, Hilbert’s weak Nullstellensatz states that if V (I) = ∅, then 1 ∈ I. Note, the
result is lost if K is not algebraically closed. The affine algebraic set VR (x2 + 1) = ∅, but
hx2 + 1i is a proper ideal in R[x].
Theorem 1.2 (Hilbert’s Strong Nullstellensatz) Let F ⊆ K be an extension of
fields where K is algebraically closed, and let I be an ideal in F[x1 , . . . , xk ]. If p ∈
√
IF (VK (I)), then p ∈ I.
Let F = (f1 , . . . , fn ) be the polynomial map of an n×n zero-nonzero pattern A . It is
worth noting that in order to prove that A is a relaxed spectrally arbitrary pattern over
F it suffices to show that VF (f1 − α1 , . . . , fn − αn ) is nonempty for all (α1 , . . . , αn ) ∈ Fn .
We say (a1 , a2 , . . . , ak ) ∈ Fk is strictly nonzero if ai 6= 0, for each i. In order to prove
that A is spectrally arbitrary over F it suffices to show that there exists some strictly
nonzero point (a1 , . . . , ak ) in VF (f1 − α1 , . . . , fn − αn ).
1.2.3
The Digraph of a Matrix
A pattern A is said to be reducible if for some integer r with 1 ≤ r ≤ n − 1, there exists
a permutation matrix P such that


PA PT = 

X
0r,n−r
Y 
.
W
A 1 × 1 nonzero block is irreducible. Otherwise, a pattern A is irreducible if it is not
reducible.
For a matrix A we define the directed graph or digraph of A, G(A) to be the vertex set
{1, 2, . . . , n} with directed arcs from vertex i to vertex j if and only if aij 6= 0. A directed
path of length q in this digraph is a sequence of q arcs (i1 , i2 ), (i3 , i4 ), . . . , (iq , iq+1 ). If
10
the vertices i1 , . . . , iq+1 are distinct, the path is called a simple path of length q. If the
vertices i1 , . . . , iq are distinct and iq+1 = i1 , then the path is called a simple cycle of
length q. A cycle of length one is called a loop.
A directed graph G is called strongly connected if for every pair of disjoint vertices
i, j there is a path from i to j and there is a path from j to i. It is well known that a
matrix A is irreducible if and only if its directed graph G(A) is strongly connected. The
characteristic polynomial of an irreducible matrix A can be easily computed from the
digraph A using the following lemma.
Lemma 1.3 Let A be a realization of an irreducible pattern A whose characteristic
polynomial is
pA (x) = xn − E1 (A)xn−1 + E2 (A)xn−2 + · · · + (−1)n En (A).
Then Ek (A) =
P
(signγk )CA (γk ) with the sum taken over all disjoint cycle unions γk of
size k in the directed graph of A , G(A ), where signγk = (−1)pk , pk is the number of
cycles of even length in γk , and CA (γk ) is the product of the entries aij on the disjoint
cycles in γk .
For an n × n matrix A over F, let S = {(i1 , j1 ), . . . , (ip , jp )} be a set of positions of A
whose entries are nonzero. We define GS to be the (non-directed) graph with vertex sex
{1, 2, . . . , n} and edge set S. The graph GS is called a tree if it is a connected, acyclic
graph that spans the vertex set {1, 2, . . . , n}. Consider the following lemma.
Lemma 1.4 Let A be an irreducible n × n matrix with m nonzero entries. Then there
exists a set S = {(i1 , j1 ), . . . , (in−1 , jn−1 )} consisting of n − 1 positions of A such that
GS is a tree of A. For any such set S, there exist nonzero numbers d1 , . . . , dn such that
the (ik , jk )-entry of D−1 AD equals 1, where D = diag(d1 , . . . , dn ).[26]
11
Let A be any realization of an irreducible n×n pattern A and since D = diag(d1 , . . . , dn )
is a diagonal matrix, D−1 AD is also a realization of A and pA (t) = pD−1 AD (t). Thus, if
A has m nonzero entries, we can consider the polynomial map F of A as a map from
Fm−(n−1) to Fn by using Lemma 1.4 to replace n − 1 off-diagonal entries of A with 1. We
primarily focus on irreducible pattern on this dissertation, so we will freely use Lemma
1.4 to reduce the number of variables in the polynomial map of the pattern.
1.3
Motivational Questions
It is shown in the paper by Drew, Johnson, Olesky, and van den Driessche[7] that if the
nilpotent-Jacobian method is used to show that a pattern is spectrally arbitrary over R
or C, then each of its superpatterns is also spectrally arbitrary over R or C, respectively.
This lead to the superpattern conjecture.
Conjecture 1.5 (The Superpattern Conjecture)
If a pattern is spectrally arbitrary over a field F, then each of its superpatterns is
spectrally arbitrary over F.
It may be reasonable to assume that the superpattern conjecture is true, because a
superpattern will yield more variables to manipulate in satisfying a given realization.
However, a superpattern could yield a drastically different combinatorial structure than
its subpattern such as possibly adding multiple k-cycles to the digraph of the superpattern. In chapter 5, we explore a number of spectrally arbitrary patterns over the
finite field F3 that have superpatterns which are not spectrally arbitrary over F3 , so the
superpattern conjecture in its most general form is false.
12
Observation 1.6 The superpattern conjecture over a general field is false.
However, attempts at discovering counterexamples to the superpattern conjecture
over larger order finite fields or field extensions of Q have so far been unfruitful. Thus,
it may be the somewhat pathological nature of F3 that yields such counterexamples, so
the following conjectures are still open.
Conjecture 1.7 If a pattern is spectrally arbitrary over F, a field extension of Q, then
each of its superpatterns is spectrally arbitrary over F.
Conjecture 1.8 If a pattern is spectrally arbitrary over Fq where q > 3, then each of
its superpatterns is spectrally arbitrary over Fq .
In chapter 2 we develop a method for determining if a pattern is spectrally arbitrary
over a field with a defined metric, such as Q and Q, called the surjective neighborhoods
condition. Unfortunately, if a pattern does satisfy the surjective neighborhoods condition
no claims about its superpatterns can be made. It may be true that each of the pattern’s
superpatterns will be spectrally arbitrary, but it is difficult to prove this generally over
non-topologically complete fields such as Q and Q. The characteristic polynomial of
a matrix depends continuously on the entries of the matrix, but this fact cannot be
utilized as powerfully over non-topologically complete fields as it can over R or C. Thus,
we would like to develop a method for determining if the superpatterns of a spectrally
arbitrary pattern remain spectrally arbitrary over an arbitrary field or an arbitrary field
extension of Q.
Question 1.9 If a pattern satisfies the surjective neighborhoods condition over F, is it
necessarily true that each of its superpatterns is spectrally arbitrary over F?
13
Question 1.10 Is there a method for examining superpatterns of spectrally arbitrary
patterns over field extensions of Q?
Question 1.11 Is there a method for examining superpatterns of spectrally arbitrary
patterns over finite fields?
For every 3 × 3 and 4 × 4 pattern explored in this paper that is spectrally arbitrary
over a some field extension F of Q, each of its superpatterns are spectrally arbitrary over
F. We speculate that this will remain true for any pattern that is spectrally arbitrary
over F, but a larger order counterexample may be discovered.
It was determined in the papers [3] and [6] that every 3 × 3 and 4 × 4 pattern that
is spectrally arbitrary over R is spectrally arbitrary over C. In fact, every pattern that
is shown to be spectrally arbitrary over R by using the nilpotent-Jacobian method will
also be spectrally arbitrary over C. It not known if this result can be generalized to any
pattern.
Conjecture 1.12 Every real spectrally arbitrary pattern will be spectrally arbitrary over
C.
The generalization of this conjecture where R ⊆ C is replaced with any extension
of fields is false. We consider a counterexample in chapter 5. However, the following
conjecture remains unproven.
Conjecture 1.13 Let Q ⊆ F ⊆ E be an extension of fields. If a pattern is spectrally
arbitrary over F, then it is spectrally arbitrary over E.
A number of patterns were explored in the paper by McDonald and Yielding [3]
that are spectrally arbitrary over C yet fail to be spectrally arbitrary over R, so the
14
conjecture that a spectrally arbitrary pattern over a general field F will be spectrally
arbitrary over any subfield of F is false. However, C and R differ greatly algebraically,
so we analyze these complex spectrally arbitrary patterns that are not real spectrally
arbitrary to determine which subfields of C they are spectrally arbitrary. This led to
the following observation.
Observation 1.14 Every 3 × 3 and 4 × 4 spectrally arbitrary pattern over R or C is
spectrally arbitrary over Q. Every 3 × 3 and 4 × 4 spectrally arbitrary pattern over R is
spectrally arbitrary over the real algebraic closure of Q.
The complex field is algebraically closed, but it also contains transcendental numbers.
If no transcendental numbers are given as coefficients of a characteristic polynomial,
would transcendental numbers be necessary in every matrix realization that satisfies the
given characteristic polynomial? In other words, will transcendental numbers ever be
necessary to solve a system of multivariate polynomials with algebraic coefficients? This
questions directly pertains to zero-nonzero matrix patterns as each component of the
polynomial map of a matrix pattern is a multivariate polynomial with integer coefficients.
It is true that transcendental numbers can solve such a system (consider f = x + y and
the solutions (π, −π) and (1, −1)), but are transcendental numbers necessary to solve
such a system? We prove in chapter 5 that transcendental numbers are in fact not needed
to solve such systems of polynomial equations. However the following conjectures remain
open.
Conjecture 1.15 Every real spectrally arbitrary pattern is spectrally arbitrary over the
real algebraic closure of Q.
15
Conjecture 1.16 If a system of multivariate polynomials with real algebraic coefficients
has a common zero, does there exist a common zero that consists of real algebraic numbers? In addition, if there exists a strictly nonzero common zero for the system, does
there exist a strictly nonzero common zero that consists of real algebraic numbers?
There are a number of real spectrally arbitrary patterns that are not spectrally
arbitrary over Q[6], but can we say anything about a rational spectrally arbitrary pattern? All known examples of rational spectrally arbitrary patterns are real spectrally
arbitrary, and we prove in chapter 5 that every rational spectrally arbitrary pattern is
relaxed spectrally arbitrary over R. However, the the following conjecture remains open.
Conjecture 1.17 Every rational spectrally arbitrary pattern is real spectrally arbitrary.
The problem of determining the lower bound of the number nonzero entries in a
spectrally arbitrary pattern was first proposed in the paper by Britz, McDonald, Olesky,
and van den Driessche[1]. The authors in this paper proved that the minimum number
of nonzero entries in a n × n spectrally arbitrary pattern over R is 2n − 1, and it was
conjectured that this bound can be strengthened to 2n. Bryan Shader proved that 2n
is the correct lower bound for spectrally arbitrary patterns over finite fields by using a
counting argument[26]. There are no known spectrally arbitrary patterns over any field
with exactly 2n − 1 nonzero entries. In chapter 5, we prove that 2n − 1 is a necessary
lower bound of nonzero entries for any spectrally arbitrary pattern over a dense subfield
of R or C. We believe, along with the authors of [1], that this bound can be strengthened
to 2n, but a general proof is still needed.
Conjecture 1.18 (The 2n Conjecture)
Every spectrally arbitrary pattern has at least 2n nonzero entries.
16
These are some of the questions that have motivated our work in this dissertation,
and we explore additional interesting questions and observations throughout this work.
17
Chapter 2
The Jacobian Method and the Surjective
Neighborhoods Condition
The nilpotent-Jacobian method was first proposed in the paper by Drew, Johnson,
Olesky, and van den Driessche[7] in 2000 as a sufficient way to check if a sign pattern is
spectrally arbitrary over R. In the same paper, it was also shown that if a sign pattern is
shown to be spectrally arbitrary over R, then each of its superpatterns is also spectrally
arbitrary over R. We can replace each instance of + and − in a spectrally arbitrary
sign pattern with ∗ to generate a spectrally arbitrary zero-nonzero pattern. Hence, the
nilpotent-Jacobian method can be used on zero-nonzero matrix patterns, and if it is used
on a zero-nonzero pattern each of its superpattern will also be spectrally arbitrary over
R. Clearly, this is a powerful tool as it proves that a (possibly) large set of patterns is
spectrally arbitrary if it is used to show a given subpattern is spectrally arbitrary. Many
authors (see [10],[1],[20],[2]) have utilized this tool to prove that a variety of patterns
and families of patterns are spectrally arbitrary over R and C. However, there are
some computational limitations to the method, as we shall explore, and it is unknown
whether it is necessary that a spectrally arbitrary zero-nonzero pattern over R satisfy
the nilpotent-Jacobian method.
Conjecture 2.1 If a pattern is spectrally arbitrary over R, then the pattern will satisfy
the nilpotent-Jacobian method.
18
In [6] the authors show that all 4×4 spectrally arbitrary zero-nonzero patterns satisfy
the nilpotent-Jacobian method, and in [10] the authors show that all 3 × 3 spectrally
arbitrary zero-nonzero patterns satisfy the nilpotent-Jacobian method. The nilpotentJacobian method is not a necessary condition for a zero-nonzero pattern to be spectrally
arbitrary over C as the authors in [3] provide two 4 × 4 spectrally arbitrary patterns over
C that do not satisfy the nilpotent-Jacobian condition. We will study these patterns in
detail in chapter 4.
Also, the nilpotent-Jacobian method can only be used over fields with a topologically
complete metric (such as R and C) as the contraction mapping theorem is utilized in
its proof. We analyze the proof of the nilpotent-Jacobian method in order to develop
a sufficient condition to determine if a zero-nonzero pattern is spectrally arbitrary over
fields with topologically incomplete metrics such as Q and algebraic extensions of Q.
2.0.1
The Proof of the nilpotent-Jacobian Method
In this section we prove that if a zero-nonzero pattern satisfies the nilpotent-Jacobian
method, then it and all of its superpatterns are spectrally arbitrary over R.
Theorem 2.2 (Nilpotent-Jacobian Method)
Let A be a n × n zero-nonzero pattern with real (complex) nilpotent realization A =
(aij ). Let U = {(i1 , j1 ), . . . , (in , jn )} be n positions of nonzero entries of A. Define D to
be the matrix obtained from A by replacing the (ik , jk )-entry by the indeterminate xk for
k = 1, . . . , n. Define a = (a1 , a2 , . . . , an ) where each ak = aik jk . Let F = (f1 , . . . , fn ) be
the polynomial map of A determined by D.
If the Jacobian of F evaluated at a is invertible, then A and all of its superpatterns
19
are spectrally arbitrary over R.
Proof. As A is nilpotent, fi (a) = 0 for each i, so F (a) = (0, . . . , 0). Since the
Jacobian of F evaluated at a is invertible the inverse function theorem asserts that there
exists open neighborhoods M and N of a and F (a) = (0, . . . , 0) respectively such that
F : M → N is a bijection. Since the inverse is continuous and each ai 6= 0, we can
pick c = (c1 , . . . , cn ) with |ci | small enough so that for b = (b1 , . . . , bn ) ∈ M such that
F (b) = c, each bi is nonzero. Hence, for each i, fi (b) = ci . Let B be the matrix obtained
from D by replacing each (ik , jk )-entry by bk . Thus, B is a realization of A and has
characteristic polynomial pB (t) = tn + c1 tn−1 + · · · + cn . Since c was chosen arbitrarily
(except for component-wise proximity to 0), we use scalar multiples of B to conclude
that A is spectrally arbitrary over R.
Let A˜ be a superpattern of A of fill p and let {(in+1 , jn+1 ), . . . , (in+p , jn+p )} be
the set of zero entries of A that are nonzero in A˜. We prove A˜ is spectrally arbitrary by showing there is a realization of A˜ whose characteristic polynomial is tn +
c1 tn−1 + · · · + cn . Let D̃ be the matrix obtained from B by replacing each (ik , jk )entry by xk for k = 1, . . . , n and by replacing each (il , jl )-entry by indeterminate yl for
l = n + 1, . . . , n + p. Let F̃ : Rn+p → Rn be the polynomial map of A˜ determined
by D̃. Note that F̃ (b1 , . . . , bn , 0, . . . , 0) = F (b1 , . . . , bn ) = (c1 , . . . , cn ), so the Jacobian
of F̃ evaluated at (b1 , . . . , bn , 0, . . . 0) equals the Jacobian of F evaluated at (b1 , . . . , bn ).
Since b = (b1 , . . . , bn ) ∈ M , the Jacobian of F evaluated at b is invertible and thus the
implicit function theorem applies to F̃ at (b1 , . . . , bn , 0, . . . , 0). Thus for (n+1 , . . . , n+p )
sufficiently close to (0, . . . , 0) there exists a unique (b̃1 , . . . , b˜n ) ∈ M such that F̃ maps
(b̃1 , . . . , b̃n , n+1 , . . . , n+p ) to (c1 , . . . , cn ). We can pick each n+l 6= 0 and let B̃ be the
20
matrix obtained from D̃ by replacing each (ik , jk )-entry by b̃k for k = 1, . . . , n and by
replacing each (il , jl )-entry by n+l for l = 1, . . . , p. Then B̃ is the desired realization of
A˜.
One of the complications in using the nilpotent-Jacobian method is knowing which
n nonzero entries to replace with indeterminates in a nilpotent realization. The authors
in [4] address this issue by noting that the nilpotent-Jacobian method can be extended
by determining the rank of the Jacobian of the polynomial map of the pattern at each
nilpotent realization.
Theorem 2.3 (Extended nilpotent-Jacobian method)
Let A be an n × n zero-nonzero pattern with polynomial map F . If there exists a
nilpotent realization A over R such that the Jacobian of F evaluated at A has rank n,
then A and each superpattern of A is spectrally arbitrary over R.
Note, the inverse function theorem and the implicit function theorem hold over C as
it is a topologically complete metric space, so the nilpotent-Jacobian method and the
extended nilpotent-Jacobian method hold over C as well.
2.0.2
Surjective Neighborhoods Condition
Some observations on the proof of the nilpotent-Jacobian method: First, note that the
inverse function theorem and implicit function theorem rely on the contraction mapping
principle, which holds over any field with a topologically complete metric space. Thus
the nilpotent-Jacobian method is sufficient in determining if a pattern (and all of its
superpatterns) is spectrally arbitrary over any such field. Second, we do not need the
21
polynomial map F determined by U to be injective in a neighborhood of a in order for
A to be spectrally arbitrary. It suffices that there exists open neighborhoods M and N
of a and F (a) = 0 respectively such that F : M → N is onto and for all (b1 , . . . , bn ) ∈ M ,
bi 6= 0.
Observation 2.4 Let F be a field with a defined metric and let A be an n × n zerononzero pattern with m nonzero entries. Let A = (aij ) be a nilpotent realization over
F and let U = {(i1 , j1 ), . . . , (in , jn )} be n positions of nonzero entries of A. Define D
to be the matrix obtained from A by replacing the (ik , jk )-entry by the indeterminate xk .
Define a = (a1 , a2 , . . . , an ) where each ak = aik jk . Let F = (f1 , . . . , fn ) be the polynomial
map of A determined by D. We say that A satisfies the surjective neighborhoods
condition if there exists open neighborhoods M and N of a and F (a) = 0, respectively,
such that F : M → N is onto and for all (b1 , . . . , bn ) ∈ M , bi 6= 0. If A satisfies the
surjective neighborhoods condition over F, then it is spectrally arbitrary over F.
Although satisfying the surjective neighborhoods condition over any field E that
has a defined metric proves that a pattern is spectrally arbitrary over E, it can be
computationally more complicated than the nilpotent-Jacobian method. The surjective neighborhoods method requires solving a system of polynomial equations, whereas
the nilpotent-Jacobian method requires computing the rank of the Jacobian. Also, we
cannot say anything about any superpattern of a pattern that satisfies the surjective
neighborhoods condition. The implicit function theorem is a powerful tool that fails
over a non-topologically complete field such as Q. We do speculate that if a pattern
satisfies the surjective neighborhoods condition then every one of its superpatterns will
satisfy it as well. This is the case for all 3 × 3 and 4 × 4 patterns studied in this paper.
22
However, a larger pattern may yield a counterexample.
Conjecture 2.5 If a pattern is shown to be spectrally arbitrary over a field F with
a defined metric by the surjective neighborhoods condition, then every superpattern is
spectrally arbitrary over F.
More generally, to prove that an n×n pattern A with m nonzero entries is spectrally
arbitrary over any field with a defined metric it suffices to show that for all (c1 , . . . , cn )
arbitrarily close to (0, . . . , 0) in Fn , there exists (b1 , . . . , bm ) with each bi 6= 0 such that
F (b1 , . . . , bm ) = (c1 , . . . , cn ), where F is the polynomial map of B.
Observation 2.6 (Extended Surjective Neighborhoods Condition) Let A be an n × n
pattern with m nonzero entries and let F : Fm → Fn be the polynomial map of A , where
F is a field with a defined metric. Let A be a nilpotent realization over F. If there exists
open neighborhoods M and N of A and (0, . . . , 0), respectively, such that F : M → N is
onto and for all (b1 , . . . , bm ) ∈ M, bi 6= 0, then A is spectrally arbitrary over F.
2.0.3
The Nilpotent-Jacobian Method over Extensions of Q
In restricted cases the nilpotent-Jacobian method can be used to prove that a pattern is
spectrally arbitrary over topologically incomplete fields such as Q and Q. To illustrate,
we consider the 3 × 3 companion-style pattern C3 .

 ∗ ∗ 0

C3 = 
 ∗ 0 ∗

∗ 0 ∗






23
The following realization is nilpotent at (x1 , x2 , x3 ) = (−1, −1, −1).


 x1 1 0 


 x 0 1 

 2


x3 0 1
The polynomial map defined by this matrix is F (x1 , x2 , x3 ) = (−x1 − 1, x1 − x2 , x2 − x3 ).
Since F has a nonsingular Jacobian evaluated at (−1, −1, −1), C3 satisfies the nilpotentJacobian method. Hence, there exists neighborhoods M and N of (−1, −1, −1) and
(0, 0, 0), respectively, such that F : M → N is onto and for all (m1 , m2 , m3 ) ∈ M ,
mi 6= 0. Let c ∈ N ∩ Q3 , so we know there is a solution to the system of equations
F (x1 , x2 , x3 ) = (c1 , c2 , c3 ), written below, in M .
−x1 − 1 = c1
(2.1)
x 1 − x 2 = c2
x 2 − x 3 = c3
We can rewrite the system of linear equations (2.1) as the matrix equation Jx = c̃,
where x = (x1 , x2 , x3 )T , the transpose of (x1 , x2 , x3 ), c̃ = (c1 + 1, c2 , c3 )T , and J is the
following matrix.

0
 −1 0

J =
 1 −1 0

0
1 −1






Note that J is the Jacobian of F , so the solution to the system of equations (2.1) is
J −1 c̃ and this solution is unique. Hence, J −1 c̃ lies in M ∩ Q3 , so F maps M ∩ Q3 onto
N ∩ Q3 . Thus C3 satisfies the surjective neighborhoods condition over Q implying that
it is spectrally arbitrary over Q.
24
We can extrapolate this argument for any 3 × 3 matrix that has a nonzero column
(or row) as follows. Let F be a field extension of Q, and let A be a nilpotent realization
at (x1 , x2 , x3 ) = (a11 , a21 , a31 ).

 x1 a12 a13

A=
 x2 a22 a23

x3 a32 a33






Let F = (f1 , f2 , f3 ) be the polynomial map determined by A, where each fi is given
below.
f1 = −x1 − (a22 + a33 )
f2 = (a22 + a33 )x1 − a12 x2 − a13 x3 + (a22 a33 − a23 a32 )
(2.2)
f3 = (a23 a32 − a22 a33 )x1 + (a12 a33 − a13 a32 )x2 + (a13 a22 − a12 a23 )x3
We assume the Jacobian of F evaluated at a = (a11 , a21 , a31 ) is nonsingular. Thus, there
exists neighborhoods M and N of a and 0, respectively, such that F : M → N is a
bijection and for all (m1 , m2 , m3 ) ∈ M , mi 6= 0. Let (c1 , c2 , c3 ) ∈ F3 with each |ci | small
enough so that (c1 , c2 , c3 ) ∈ N .
We set fi = ci for i = 1, 2, 3 and rewrite the system of equations (2.2) as follows:
−x1 = c1 + γ1
β21 x1 + β22 x2 + β23 x3 = c2 + γ2
(2.3)
β31 x1 + β32 x2 + β33 x3 = c3 + γ3
Since each γi and βji is a polynomial combination of aij , they all lie in F (for example,
γ2 = a23 a32 − a22 a33 , γ3 = 0, and β21 = −a12 ). We can rewrite the system of linear
equations (2.3) as the matrix equation Jx = c̃, where x = (x1 , x2 , x3 )T , c̃ = (c1 + γ1 , c2 +
25
γ2 , c3 + γ3 )T , and J is the following matrix.

0
 −1 0

J =
 β21 β22 β23

β31 β32 β33






Since J is the Jacobian of F at a, J is invertible and J −1 ∈ F3×3 . Hence, J −1 c̃ ∈ F3
is the unique solution to the system, implying J −1 c̃ ∈ M . Thus, F maps M ∩ F3 onto
N ∩ F3 .
If a polynomial map F is obtained from a nilpotent realization of an n × n pattern
where n nonzero entries in one column (or row) are changed to indeterminates, then the
polynomial map will be linear. Thus, for c ∈ Fn any vector b that solves the system
F (b) = c also lies in Fn . So, we can show that the n × n companion-style pattern Cn
(shown below) and all of its superpatterns are spectrally arbitrary over F for all fields
such that Q ⊆ F ⊆ C by applying the following theorem.


∗ ∗ 0
· 0




 ∗ 0 ∗ ... 0 






Cn =  ∗ 0 . . . . . . 0 



 . . .
 .. .. . . 0 ∗ 




∗ 0 ··· 0 ∗
Theorem 2.7 Let Q ⊆ F ⊆ C be an extension of fields, and let A be a n × n pattern
that has at least one strictly nonzero column (or row) U = {(i1 , j1 ), . . . , (in , jn )}. Assume
A has a nilpotent realization A ∈ Fn×n . Replace the nonzero column (or row) in A with
n indeterminates, and let F be the polynomial map of A determined by this matrix. If
the Jacobian of F evaluated at a = (ai1 ,j1 , . . . , ain ,jn ) is nonsingular, then A and all of
its superpatterns are spectrally arbitrary over F.
26
Proof. Without loss of generality, we assume that the first column of A is strictly
nonzero. Let A = (aij ) ∈ Fn×n be a nilpotent realization of A , and we define D by
replacing the first column of A by n indeterminates.


x1 a12 · · ·




D=




x2 a22
..
..
.
.
xn an2
a1n 


· · · a2n 

.. 
...
. 


· · · ann
Let F = (f1 , . . . , fn ) be the polynomial map determined by D, where each fi is given
below.
f1 = −x1 − γ1
f2 = β21 x1 + β22 x2 + · · · + β2n xn − γ2
..
.
(2.4)
fn = βn1 x1 + βn2 x2 + · · · + βnn xn − γn
Note that each fi is linear as the indeterminates lie on the same column. Since each γi
and βji are polynomial combination of the aij , each γi and βji lies in F. By assumption
the Jacobian of F evaluated at a = (a11 , a21 , . . . , an1 ) is nonsingular, so there exists
neighborhoods M and N in Cn of a and 0, respectively, such that F : M → N is a
bijection and for all (m1 , m2 , . . . , mn ) ∈ M , mi 6= 0.
We set each fi = ci , where ci ∈ F and each |ci | is small enough so that (c1 , c2 , . . . , cn ) ∈
N and rewrite the system of equations (2.4) into the matrix equation Jx = c̃, where
27
x = (x1 , . . . , xn )T , c̃ = (c1 + γ1 , . . . , cn + γn )T ,

 −1 0


 β21 β22
J =
 .
..
 ..
.


βn1 βn2
and J is the following matrix

0
0 


· · · β2n 

.. 
..
.
. 


· · · βnn
Note that J is the Jacobian of F at a, so J is invertible and J −1 ∈ Fn×n . Hence,
J −1 c̃ ∈ Fn is the unique solution to the system, implying J −1 c̃ ∈ M . Thus, F : M ∩Fn →
N ∩ Fn is onto, so A satisfies the surjective neighborhoods condition over F.
Let A˜ be a superpattern of A of fill p and let {(i1 , j1 ), . . . , (ip , jp )} be the set of
zero entries of A that are nonzero in A˜. Let B = (bij ) be the realization of A
over F whose characteristic polynomial is pB (t) = tn + c1 tn−1 + · · · + cn . We assume (b11 , b21 , . . . , bn1 ) ∈ M . Let D̃ be the matrix obtained from B by replacing
each (k, 1)-entry by xk for k = 1, . . . , n and by replacing each (il , jl )-entry by inde
terminate yl for l = 1, . . . , p. Let F̃ = f˜1 , . . . , f˜n : Fn+p → Fn be the polynomial map of A˜ determined by D̃. Since the indeterminates xi all lie on one column,
each coefficient function f˜i has the form f˜i = δi1 x1 + · · · + δin xn + γ̃i , where δij ∈
F[y1 , . . . , yp ] and γ̃i ∈ F for i = 1, . . . , n and j = 1, . . . , n. Since (b11 , b21 , . . . , bn1 ) ∈ M
and F̃ (b11 , b21 , . . . , bn1 , 0, . . . , 0) = F (b11 , b21 , . . . , bn1 ), the Jacobian of F̃ evaluated at
(b11 , b21 , . . . , bn1 , 1 , . . . , p ) is invertible for (1 , . . . , p ) close enough to (0, . . . , 0). But
the Jacobian of F̃ evaluated at (b11 , b21 , . . . , bn1 , 1 , . . . , p ) is the n × n matrix J whose
(i, j)-entries are δij (1 , . . . , p ). Thus, J −1 (c1 − γ̃1 , . . . , cn − γ̃n )T is the unique solution
to the system of equations, F̃ (x1 , . . . , xn ) = (c1 , . . . , cn ), and for all (1 , . . . , p ) in Fn sufficiently close to (0, . . . , 0), J −1 (c1 − γ̃1 , . . . , cn − γ̃n )T lies in M ∩ Fn . Hence, A˜ satisfies
the surjective neighborhoods condition over F.
28
Chapter 3
Small Patterns
In this chapter, we classify all 2 × 2 and 3 × 3 irreducible patterns that are spectrally
arbitrary over some extension of Q. For the remainder of this chapter, we let F denote
an arbitrary field extension of Q.
3.1
2 × 2 Patterns
The full 2 × 2 pattern B2 was shown to be spectrally arbitrary over R[7], C[3], and
all finite fields[18] of cardinality 5 or greater. It was shown in [18] that no irreducible
subpattern of B2 is potentially nilpotent, so the full 2 × 2 pattern is combinatorially
minimally spectrally arbitrary.
Claim 3.1 The full 2 × 2 pattern B2 is spectrally arbitrary over every field extension
of Q.
Proof. Consider the polynomial p(t) = t2 + α1 t + α2 in F[t]. Pick q ∈ F such that
α21
4
− α2 −
q2
4
6= 0, and q 6= ±α1 . Then the matrix

α21
α1 +q
− α2 −
 − 2
4
B=
q−α1
1
2
q2
4



has characteristic polynomial p(t) and each entry is nonzero and lies in F.
29
3.2
3 × 3 Patterns
In [1] and [7] all 3 × 3 spectrally arbitrary sign patterns over R were classified. Their results were used in the papers by Corpuz and McDonald[6] and McDonald and Yielding[3]
to classify all spectrally arbitrary 3 × 3 zero-nonzero patterns over R and C, respectively.
We summarize their results in the following theorem.
Theorem 3.2 Let A be an irreducible 3 × 3 pattern.
1. If A has 5 nonzero entries, then it is not potentially nilpotent over any field.
2. If A has 6 nonzero entries and is spectrally arbitrary over R, then it is permutationally similar to one of these two patterns:




 ∗ ∗ 0 
 ∗ ∗ 0 





 ∗ 0 ∗ 
C
=
T3 = 
3
∗
0
∗








∗ 0 ∗
0 ∗ ∗
Every superpattern of these two patterns is spectrally arbitrary over R.
3. The following pattern is spectrally arbitrary over C, but it is not potentially nilpotent over R.


 ∗ ∗ 0 



D3 = 
 0 ∗ ∗ 


∗ 0 ∗
Every superpattern of D3 is spectrally arbitrary over C.
4. If A has at least seven nonzero entries and at least two of these entries lie on the
diagonal, then it is are spectrally arbitrary over R and C.
30
We showed in chapter 2 that the companion-style matrix C3 and all of its superpatterns are spectrally arbitrary over every field extension of Q. We now show that the
same is true for T3 .
Claim 3.3 The 3 × 3 tridiagonal pattern T3 and all of its superpatterns are spectrally
arbitrary over every field extension of Q.
Proof. Let (q1 , q2 , q3 ) ∈ F3 . We set the polynomial map F of F3 equal to (q1 , q2 , q3 )
and solve to get the following system of equations.
x1 = −x4 − q1
x2 = −
−x34 − x24 q1 − x4 q2 − q3
+ x4 (−x4 − q1 ) − q2
2x4 + q1
−x34 − x24 q1 − x4 q2 − q3
x3 =
2x4 + q1
We can pick x4 ∈ F large enough such that x1 , x2 , x3 ∈ F are nonzero. Thus, T3 is
spectrally arbitrary over F.
There are two non-permutationally similar fill 1 superpatterns of T3 : the superpattern with the (1, 3)-entry made nonzero and the superpattern with the (2, 2)-entry
made nonzero. The following are nilpotent realizations of the fill 1 superpatterns when
(x1 , x2 , x3 ) = −1, 1, 23 and (y1 , y2 , y3 ) = − 38 , 1, 1 .


 x1 x2 x3 


 −2 0 1 




0 1 1


 −2 1 0 


 y

y
y
2
3 
 1


1
0 −3 1
Both polynomial maps determined by the above matrices have nonsingular Jacobians
at (x1 , x2 , x3 ) = −1, 1, 23 and (y1 , y2 , y3 ) = − 38 , 1, 1 , respectively, so by theorem 2.7
31
both patterns and all of their superpatterns are spectrally arbitrary over F.
The pattern D3 is not potentially nilpotent over Q, so it is not spectrally arbitrary
over Q. It is spectrally arbitrary over certain algebraic extensions of Q.
Claim 3.4 D3 is spectrally arbitrary over Q (and Q√ , in particular). Every superpattern
of D3 is spectrally arbitrary over every extension of Q.
Proof. We set the polynomial map F (x1 , x2 , x3 , x4 ) of D3 equal to (q1 , q2 , q3 ) and
solve to get the following system of equations.
q
1
2
2
x2 =
± −2q1 x4 + q1 − 4q2 − 3x4 − q1 − x4
2
x1 = −x2 − x4 − q1
x3 = −x1 x2 x4 − q3
If each qi lies in Q (or Q√ ), then we can choose x4 in Q (or Q√ ) large enough such that
x1 , x2 , x3 are all nonzero and lie in Q (or Q√ ).
Every fill 1 superpattern of D3 permutationlly similar to the following pattern.


 ∗ ∗ ∗ 


 0 ∗ ∗ 




∗ 0 ∗
The following matrix is a nilpotent realization of the fill 1 superpattern when (x1 , x2 , x3 ) =
(−2, 1, 3).


 x1 x2 x3 


 0 1 1 




−1 0 1
32
The polynomial map determined by the matrix above has nonsingular Jacobian evaluated at (x1 , x2 , x3 ) = (−2, 1, 3), so by Theorem 2.7 all superpatterns of D3 are spectrally
arbitrary over F.
We summarize all 3×3 patterns that are spectrally arbitrary over some field extension
of Q in the following theorem.
Theorem 3.5 Let A be an irreducible 3 × 3 pattern.
1. If A is spectrally arbitrary over every field extension of Q and has 6 nonzero
entries, then it is permutationally similar to one of these two patterns:




 ∗ ∗ 0 
 ∗ ∗ 0 





 ∗ 0 ∗ 
C
=
T3 = 
3
∗
0
∗








∗ 0 ∗
0 ∗ ∗
Every superpattern of these two patterns is spectrally arbitrary over all field extensions of Q.
2. The following pattern is spectrally arbitrary over Q (and Q√ , in particular).


 ∗ ∗ 0 



D3 = 
0
∗
∗




∗ 0 ∗
Every superpattern of D3 is spectrally arbitrary over every field extension of Q.
3. If A has at least seven nonzero entries and at least two of these entries lie on the
diagonal, then it is spectrally arbitrary over every field extension of Q.
33
Chapter 4
4 × 4 Patterns
In this chapter, we analyze all 4 × 4 irreducible patterns that are minimally spectrally
arbitrary over R or C. We are particularly interested in answering the following questions.
1. Is the pattern spectrally arbitrary over Q?
2. If the pattern is not spectrally arbitrary over Q, over which field extensions of Q
is the pattern spectrally arbitrary?
3. If the pattern is spectrally arbitrary over R, is it spectrally arbitrary over the real
algebraic closure of Q.
4. Is the pattern is spectrally arbitrary over Q?
In the paper by Corpuz and McDonald[6], the authors characterize all the irreducible
4×4 zero-nonzero patterns that are spectrally arbitrary over R. In the paper by Yielding
and McDonald[3], the authors characterize all the irreducible 4×4 zero-nonzero patterns
over C, and in doing so showed that every pattern that is 4 × 4 spectrally arbitrary over
R is also spectrally arbitrary over C. We will show in chapter 5 (corollary 5.7) that
every complex spectrally arbitrary pattern is spectrally arbitrary over Q. Thus, we can
answer question 4 affirmatively. We will also show in this chapter that each minimally
spectrally arbitrary patten over R is also spectrally arbitrary over the real algebraic
closure of Q.
34
Observation 4.1 Every 4×4 pattern that is spectrally arbitrary over R or C is spectrally
arbitrary over Q.
Observation 4.2 Every 4 × 4 pattern that is minimally spectrally arbitrary over R is
spectrally arbitrary over the real algebraic closure of Q.
4.1
Real Spectrally Arbitrary Patterns
4.1.1
Real Spectrally Arbitrary Patterns with 8 Nonzero Entries
Every real spectrally arbitrary pattern with 8 nonzero entries is permutationally similiar
to one of the patterns listed below. The first group includes every pattern that is
spectrally arbitrary over every field extension of Q, including Q.
The final three patterns are not spectrally arbitrary over Q, but they are spectrally
arbitrary over the real algebraic closure of Q. Two of these final three patterns are
spectrally arbitrary over the field extension Q√+ of Q. Note, Q√+ was defined in section
1.2.2.
1. Each of these patterns satisfies the surjective neighborhoods condition over every
field extension of Q (see appendix A).



 ∗ ∗ 0 0  ∗ ∗ 0 0 






 ∗ 0 ∗ ∗  ∗ 0 ∗ 0 






 0 0 ∗ ∗  0 0 ∗ ∗ 






∗ 0 0 0
∗ ∗ 0 0


∗ ∗ 0 0 


∗ 0 ∗ 0 



0 0 ∗ ∗ 


∗ 0 ∗ 0
∗ ∗ 0 0 


∗ 0 ∗ 0 


0 0 0 ∗ 


∗ ∗ 0 ∗
35

 ∗ ∗ 0


 ∗ 0 ∗


 0 0 0


∗ 0 ∗




0 


0 



∗ 


∗
∗ ∗ 0 ∗ 


∗ 0 ∗ 0 



0 0 ∗ ∗ 


0 ∗ 0 0

 ∗ 0 0


 ∗ 0 ∗


 0 0 ∗


0 ∗ 0
∗ ∗ 0 0 


∗ 0 ∗ ∗ 



0 0 ∗ ∗ 


0 ∗ 0 0

∗ 


∗ 


∗ 


∗
0 ∗ 0 ∗ 


∗ 0 ∗ 0 


0 0 ∗ ∗ 


0 ∗ 0 ∗
2. Consider the following pattern


 ∗ ∗ 0 0 




 ∗ 0 ∗ 0 




 0 ∗ 0 ∗ 




0 0 ∗ ∗
Since
with nilpotent realization.

1
0
 −1

√

0
1
 1+ 2


√ 2

0
− 1+ 2
0


√
0
0
1+ 2

0 


0 


1 


1
√
2 is needed in every nilpotent realization this pattern is not spectrally
arbitrary over Q, but it is spectrally arbitrary over Q√+ . Consider the following
realization.

 v 1


 w 0


 0 x


0 0

0 0 


1 0 


0 1 


y z
By setting the polynomial map F (v, w, x, y, z) determined by the above matrix
equal to (q1 , q2 , q3 , q4 ) where each qi ∈ Q√+ and solving, we get the following
36
system of equations.
v = −z − q1
1
√
2q1 z 2 ± γ − q1 q2 + q3 + 2z 3
2 (q1 + z)
−q1 w − 2q1 z 2 − q12 z − q2 z − q1 q2 + q3 − 2wz − z 3
x=
z
w=
y = −q1 z − q2 − w − x − z 2
Where γ is the following:
γ = −2q1 z 2 + q1 q2 − q3 − 2z 3
2
− 4 (q1 + z) (−3q1 z 4 − 3q12 z 3
− q2 z 3 − q13 z 2 − 2q1 q2 z 2 + q3 z 2 − q12 q2 z + q1 q3 z − q4 z − z 5 )
In order to ensure that v, w, x, y, z are nonzero the expression δ, given below, must
be nonzero.
δ =3q1 wz + q12 w + 3q1 z 3 + 3q12 z 2 + q2 z 2 + q13 z + 2q1 q2 z − q3 z
+ q12 q2 − q1 q3 + q4 + 2wz 2 + z 4
We can pick z ∈ Q√+ large enough such that v, w, x, y are all nonzero, γ > 0,
δ 6= 0, and v, x, y ∈ Q√+ . Since Q√+ contains the square roots of all of its positive
elements, we have w ∈ Q√+ . Hence, this pattern is spectrally arbitrary over Q√+
3. The following pattern is not spectrally arbitrary over Q or any finite field extension
of Q. It is spectrally arbitrary over Q√+ and the real algebraic closure of Q (See
appendix A).
37

 ∗ ∗ 0


 ∗ 0 ∗


 0 0 ∗


0 ∗ ∗

0 


0 


∗ 


0
4. The following pattern is spectrally arbitrary over the real algebraic closure of Q.
Consider the following pattern and nilpotent realization.


 ∗ ∗ 0 0 




 ∗ 0 ∗ 0 




 0 0 ∗ ∗ 




0 ∗ 0 ∗


1
1
0



0
1
 −a2 − a − 1



0
0
−a − 1


0
2a2 + 2a + 1
0
0 


q
0 
√
, a = 1
3+2 5−1

2

1 

a
p
√
Since a =
3 + 2 5 − 1 is necessary in any nilpotent realization, this pattern
1
2
is not spectrally arbitrary over Q.
Consider the following realization.

 w 1 0


 x 0 1


 0 0 y


0 z 0

0 


0 


1 


a
38
It was shown in [6] that the Jacobian of the polynomial map determined by the
realization above, F = (−a−w−y, aw+ay +wy −x, −awy +ax+xy −z, wz −axy),
is invertible at (2, −a2 − a − 1, −a − 1, 2a2 + 2a + 1), so there exists neighborhoods
M and N of (2, −a2 − a − 1, −a − 1, 2a2 + 2a + 1) and (0, 0, 0, 0), respectively,
such that F : M → N is a bijection and for all (m1 , m2 , m3 , m4 ) ∈ M , mi 6= 0.
Pick (q1 , q2 , q3 , q4 ) ∈ R4 ∩ N such that each qi is algebraic. Solving F (w, x, y, z) =
(q1 , q2 , q3 , q4 ), we get the following system of equations.
z = −a3 − a2 q1 − a2 y − aq1 y − aq2 − ay 2 − q1 y 2 − q2 y − q3 − y 3
w = −a − q1 − y
x = −a2 − aq1 − ay − q1 y − q2 − y 2
0 = a4 + 2a3 q1 + 3a3 y + 4a2 q1 y + a2 q12 + a2 q2 + 3a2 y 2 + 4aq1 y 2 +
+ aq12 y + 3aq2 y + aq1 q2 + aq3 + 3ay 3 + 2q1 y 3 + q12 y 2 + q2 y 2 +
+ q1 q2 y + q3 y + q1 q3 + y 4 − q4
We know there exists a unique strictly nonzero solution (w, x, y, z) to this system
in M . Since the fourth equation in the system is a single variate polynomial
with respect to y whose coefficients are all real algebraic numbers, y ∈ R is also
algebraic. Thus, w, x, z are also real algebraic numbers, implying this pattern is
spectrally arbitrary over the real algebraic numbers.
39
4.1.2
Real Spectrally Arbitrary Patterns with 9 Nonzero Entries
Every real spectrally arbitrary pattern with 9 nonzero entries is permutationally similar
to one of the patterns listed below. The first group includes every pattern that is
spectrally arbitrary every field extension of Q, including Q. The next three patterns are
spectrally arbitrary over Q√+ , and the last pattern is spectrally arbitrary over Qβ (see
section 1.2.2). Note, all patterns are spectrally arbitrary over the real algebraic closure
of Q.
1. Each of these patterns is spectrally arbitrary over every field extension of Q (see
appendix A). Every superpattern of the last two patterns below (labeled with a
diamond) is spectrally arbitrary over every field extension of Q.

 ∗ ∗ 0


 ∗ ∗ ∗


 0 0 ∗


∗ 0 0

 ∗ ∗ 0


 ∗ 0 ∗


 0 ∗ 0


∗ 0 0


∗ 


0 



∗ 


0

0 


∗ 



∗ 


∗
0 ∗ ∗ 0 


∗ 0 ∗ 0 


∗ 0 ∗ ∗ 


∗ 0 0 ∗

0 ∗ ∗ 0 


∗ 0 ∗ 0 


0 0 ∗ ∗ 


∗ ∗ 0 ∗

 ∗ ∗ ∗


 ∗ 0 ∗


 0 0 ∗


∗ 0 0

 0 ∗ 0


 ∗ 0 ∗


 ∗ 0 ∗


∗ 0 0


0 


0 



∗ 


∗

0 


∗ 



∗ 


∗
∗ ∗ 0 0 


∗ 0 ∗ 0 


0 ∗ ∗ ∗ 


∗ 0 0 ∗

∗ ∗ 0 0 


0 ∗ ∗ ∗ 


∗ 0 0 ∗ 


∗ ∗ 0 0
40

 0 ∗ 0


 0 ∗ ∗


 ∗ 0 0


∗ ∗ 0


0 


∗ 



∗ 


∗

0 ∗ ∗ ∗ 


∗ ∗ 0 0 


∗ 0 ∗ 0 


∗ 0 0 ∗

∗ ∗ ∗ 0 


∗ ∗ ∗ ∗ 


0 0 0 ∗ 


∗ 0 0 0










 0 ∗ 0


 ∗ 0 ∗


 0 0 ∗


0 ∗ ∗

 ∗ ∗ 0


 ∗ ∗ ∗


 0 0 0


0 ∗ ∗


0 


∗ 



∗ 


∗

0 


∗ 


∗ 


0
∗ ∗ 0 0 


∗ ∗ ∗ 0 


0 ∗ ∗ ∗ 


0 0 ∗ 0
2. The following patterns are spectrally arbitrary over Q√+ (see appendix A).

 0 ∗ 0


 ∗ ∗ ∗


 0 0 ∗


0 ∗ ∗



0 


0 



∗ 


∗
∗ ∗ 0 0 


0 ∗ ∗ ∗ 



0 0 ∗ ∗ 


∗ ∗ 0 0
∗ ∗ 0 0 


0 ∗ ∗ ∗ 


0 0 0 ∗ 


∗ ∗ 0 ∗
3. The following pattern is spectrally arbitrary over Qβ (see appendix A).

 ∗ ∗ 0


 ∗ ∗ ∗


 0 0 ∗


∗ 0 0

0 


0 


∗ 


∗
41
4.2
Minimally Spectrally Arbitrary Patterns over C
That Are Not Spectrally Arbitrary Over R
There are three non-permutationally similar patterns that are minimally spectrally arbitrary over C and are not spectrally arbitrary over R. Given below, these pattern were
originally studied over C in the paper by




 ∗ ∗ ∗ 0 







 ∗ ∗ ∗ 0 
 N4 = 
M4 = 




 0 0 0 ∗ 






∗ ∗ 0 0
McDonald and Yielding[3].


∗ ∗ 0 0 
 ∗ ∗




0 0 ∗ ∗ 
 ∗ 0
 B4 = 


 0 0
0 0 ∗ ∗ 




∗ 0
∗ ∗ 0 0

0 0 


∗ 0 


∗ ∗ 


0 ∗
The two patterns M4 and N4 are particularly important as they are the only known
examples of patterns that do not satisfy the nilpotent-Jacobian method, yet are spectrally arbitrary over C[3]. We analyze M4 along with all of its superpatterns in detail in
the last four sections of this chapter in order to gain insight into some of the pathological
properties of the pattern.
Claim 4.3 The pattern N4 is spectrally arbitrary over Q√ . The pattern B4 is spectrally
arbitrary over Qβ .
As with M4 , N4 is an example of a pattern that is spectrally arbitrary over C yet fails
the nilpotent-Jacobian method. However, the following nilpotent realization of N4 at
(x1 , x2 , x3 , x4 ) = (−1, −1, −1, −1) yields a polynomial map that satisfies the surjective
42
neighborhoods condition over Q√ (see [3], chapter 3).


 x1 x2 0 0 




 0 0 x3 x4 




 0 0 1 1 




1 1 0 0
The following nilpotent realization of B4 at (x1 , x2 , x3 , x4 ) = (−1−i, −i, i, −1) yields
a polynomial map that satisfies the surjective neighborhoods condition over Qβ (see
appendix A).


 x1


 x2


 0


x4
4.2.1
1
0
0
1
0 x3
0
0
0 


0 


1 


1
M4 and the Surjective Neighborhoods Condition
Since M4 is spectrally arbitrary over C we can infer that it is spectrally arbitrary over Q
by applying theorem 5.6, which we prove in chapter 5. Here we will utilize the surjective
neighborhoods condition to determine which field extensions of Q over which M4 is
spectrally arbitrary. We also show that every superpattern of M4 is spectrally arbitrary
over every field extension of Q.
Assertion 4.4 M4 satisfies the surjective neighborhoods condition over Q√
43
Proof. The following matrix is a nilpotent realization of M4 :


 −1 1 1 0 




 −1 1 1 0 




 0 0 0 1 




−1 1 0 0
Changing four nonzero entries to indeterminates, we get:


 x1 x2 x3 0 




x
1
1
0

 4


M4 = 

 0 0 0 1 




−1 1 0 0
The map F : Q4√ → Q4√ defined by
(x1 , x2 , x3 , x4 ) 7−→ (−x1 − 1, x1 − x2 x4 , x3 − 1, x1 + x2 − x3 x4 − x3 )
is the polynomial map of M4 given by M4 . We want to find a neighborhood M of a =
(−1, 1, 1, −1) and a neighborhood N of (0, 0, 0, 0) such that F : M → N is onto and every
n
o
element in M is strictly nonzero. Set N = (n1 , n1 , n3 , n4 ) : |ni | < 41 and ni ∈ Q√
and fix (c1 , c2 , c3 , c4 ) ∈ N . Setting F (x1 , x2 , x3 , x4 ) = (c1 , c2 , c3 , c4 ) and solving for
x1 , x2 , x3 , x4 we get the following system of equations:
x1 = −c1 − 1
x 3 = 1 + c3
p
c1 + c3 + c4 ± (−c1 − c3 − c4 − 2) 2 − 4 (−c1 − c2 − 1) (−c3 − 1) + 2
x4 =
2 (−c3 − 1)
−c1 − c2 − 1
x2 =
x4
(4.1)
44
Clearly, x1 6= 0, x3 6= 0, and −c3 − 1 6= 0 since |ci | < 41 . To ensure that x4 6= 0 and
x2 6= 0, we must have and c2 6= −c1 − 1. If c2 = −c1 − 1, then we get a contradiction as
1
4
> |c2 | = | − 1 − c1 | ≥ |1 − |c1 || > |1 − 41 | = 34 .
Let M be the preimage of N under F , so M ⊆ Q√ . Since F is a continuous map
and (0, 0, 0, 0) ∈ N , M is a neighborhood of a. Thus, M4 satisfies the surjective neighborhoods condition over Q√ .
Note, the polynomial map F given in the previous example does not satisfy the
surjective neighborhoods condition over R. We know this to be true as M4 is not spectrally arbitrary over R[6], but we examine exactly why F does not satisfy the surjective
neighborhoods condition.
Suppose F maps onto a real open neighborhood N of (0, 0, 0, 0). We show that
components of the preimage of N under F must contain complex numbers. We have
(0, α, 0, 0) ∈ N for some positive real number α. Setting (c1 , c2 , c3 , c4 ) = (0, α, 0, 0)
√
and solving the system of equations 4.1, we get x1 = −1, x3 = 1, x4 = 2 ± i α, and
x2 =
−2
4+α
±
√
i α
.
4+α
Of course we could examine a different polynomial map of M4 by changing the
placement of the indeterminates. Consider the following matrix:


 −1 1 x1 0 




 −1 1 1 0 




 0 0 0 x2 




x3 x4 0 0
The polynomial map determined by the matrix above is given by:
(x1 , x2 , x3 , x4 ) 7−→ (0, 0, −x1 x2 x3 − x2 x4 , x1 x2 x3 − x2 x3 + x1 x2 x4 − x2 x4 )
45
Notice that this function cannot map onto any neighborhood of the origin. By changing
the placement of the indeterminates, there are 94 = 126 possible polynomial maps
that can be created from this one nilpotent realization, and there are other nilpotent
realizations of M4 to consider. Thus, it is not computationally feasible to use the
surjective neighborhoods property to check to see if a pattern is not spectrally arbitrary
over a specific field.
4.2.2
M4 Over Q and Extensions of Q
In [8], the authors noted that M4 cannot satisfy the polynomial p(t) = t4 + t2 . We
analyze the pattern in more detail to determine exactly which polynomials it fails to
satisfy over Q and R. We also give the smallest (with respect to set inclusion) field
extension of Q over which M4 is spectrally arbitrary.
Assertion 4.5 Over Q (and R), M4 satisfies all polynomials of the form p(t) = t4 +
α1 t3 + α2 t2 + α3 t + α4 where α3 6= 0 or α4 6= 0. Over Q (and R), M4 satisfies no
polynomials of the form p(t) = t4 + α1 t3 + α2 t2 + α3 t + α4 where α3 = 0, α4 = 0, and
α12 − 4α2 < 0.
Proof. Assume α3 6= 0 or α4 6= 0 and consider

 a 1 1


 b c d
M =

 0 0 0


e f 0
the following realization.

0 


0 


1 


0
46
Setting the characteristic polynomial pM (t) equal to p(t) and solving, we get the following
system of equations:
a = α1 − c
b = −c2 − cα1 − α2
(4.2)
e = −df − α3
cα3 − dα3 + α4
− 2dc + α1 c + d2 − dα1 + α2
Substituting a, b, e, f back into M , we get

−c − α1
1



−c2 − α1 c − α2
c


M =

0
0


d(cα3 −dα3 )
cα3 −dα3 +α4
−α3 − c2 −2dc+α
2
c2 −2dc+α1 c+d2 −dα1 +α2
1 c+d −dα1 +α2
f=
c2

1 0 


d 0 


0 1 


0 0
To get a matrix realization of M4 with characteristic polynomial p(t) such that each
appropriate entry is a nonzero rational number we must have c not equal any of the
following numbers:
(i) −α1
(ii)
± 12
−α1 −
(iii)
1
2
(iv)
dα3 −α4
α3
(v)
p
2d − α1 ±
α12
− 4α2 .
p
α12 − 4α2
√ √
± α3 d2 α3 +2dα1 α3 −4dα4 +α21 α3 −4α2 α3 +dα3 −α1 α3
2α3
We can pick nonzero c, d ∈ Q so that c will not equal any of the numbers listed
above.
47
If α3 = 0 and α4 = 0, then we get the following system of equations by setting
p(t) = pM (t).
a = α1 − c
b = −c2 − cα1 − α2
(4.3)
e = −df − α3
q
1
2
2d − α1 ± α1 − 4α2
c=
2
If α12 − 4α2 < 0, then c cannot be a real number. Note that if α12 − 4α2 > 0, d
and f can be chosen large enough to make a, b, c, e nonzero. Thus, M4 will satisfy all
polynomials of the form p(t) = t4 + α1 t3 + α2 t2 + α3 t + α4 over R so long as α3 6= 0,
α4 6= 0, or α3 = 0 and α4 = 0 with α12 − 4α2 > 0.
Claim 4.6 The field Q√ is the smallest field extension of Q over which M4 is spectrally
arbitrary.
Proof. Let M and p(t) be defined as in the proof of assertion (4.5). If α3 6= 0 or
α4 6= 0, then by solving the system of equations 4.2, p(t) is realized by any F where
F ⊇ Q. If α3 = 0 and α4 = 0, we consider the system of equations (4.3). By the
definition of Q√ , c ∈ Q√ for any choice of α1 , α2 ∈ Q√ . Thus, a and b also lie in
Q√ . We can pick d, f ∈ Q large enough so that c, a, b, e are all nonzero. Hence, M4 is
spectrally arbitrary over Q√ .
To prove Q√ is the smallest field extension over which M4 is spectrally, we first show
that M4 is not spectrally arbitrary over any finite dimensional field extension of Q, and
then we show that M is not spectrally arbitrary over any strict subset of Q√ . Let F be
a finite dimensional field extension of Q, so there is a prime number p ∈ Q such that
48
√
√
p∈
/ F. Solving the system of equations pM (t) = t4 − p4 t2 for c yields c = d ± p ∈
/ F, so
M4 is not spectrally arbitrary over F. Let Q 1n be a proper subset of Q√ . By construc2
√
√
√
2n
n
n+1
tion, 2 2 ∈ Q 1n but 2 2 ∈
/ Q 1n . Solving the system of equations pM (t) = t4 − 4 2 t2
2
2
p √
√
n+1
2n
2 = d± 2 2 ∈
/ Qn . Thus, M4 is not spectrally arbitrary over
for c yields c = d ±
Qn .
4.2.3
Fill 1 Superpatterns of M4
We assume F is an arbitrary field extension of Q for the remainder of this chapter.
Every fill 1 superpattern of M4 is permutationally similar to one of the following three
patterns. We will show that each fill 1 superpattern satisfies the surjective neighborhoods
condition over F.

 ∗ ∗ ∗


 ∗ ∗ ∗


 0 0 ∗


∗ ∗ 0
 
0 


0 
,

∗ 


0
 ∗ ∗ ∗


 ∗ ∗ ∗


 0 0 0


∗ ∗ 0
 
∗ 


0 
,

∗ 


0
 ∗ ∗ ∗


 ∗ ∗ ∗


 0 0 0


∗ ∗ ∗

0 


0 


∗ 


0
Claim 4.7 The fill 1 superpattern of M4 with the (3, 3)-entry made nonzero satisfies
the surjective neighborhoods condition over F.
Proof. Consider the following nilpotent realization.
49


1
1 0 
 4




−21
−5
−6
0







 0
0
1
1




1
0
0
1
3
Changing four nonzero entries to indeterminants, we get:


1 0 
 x1 1




 x2 −5 −6 0 




 0 0
1 1 




x3 x4 0 0
Set a = 4, −21, 1, 13 , and the polynomial map F : F4 → F4 is given by:
(x1 , x2 , x3 , x4 ) 7−→ (4 − x1 , −4x1 − x2 − 5, 5x1 + x2 − x3 + 6x4 , x3 − 6x1 x4 − x2 x4 )
Setting F (x1 , x2 , x3 , x4 ) = (c1 , c2 , c3 , c4 ) ∈ F4 and solving for each xi , we get the following
system of equations:
x 1 = 4 − c1
x2 = 4c1 − c2 − 21
x4 =
c1 + c2 + c3 + c4 + 1
2c1 + c2 + 3
x3 = −c1 − c2 − c3 + 6x4 − 1
For each i we can fix |ci | < for sufficiently small so that each xi will be nonzero
and the denominator of x4 will be nonzero. Set N = {(n1 , n2 , n3 , n4 ) ∈ F4 : |ni | < },
and let M be the preimage of N under F . Since F is a continuous map, M is an open
neighborhood of a in F4 , and for all (m1 , m2 , m3 , m4 ) ∈ M , mi 6= 0. Thus, the pattern
50
satisfies surjective neighborhoods condition is satisfied over F.
Claim 4.8 The fill 1 superpattern of M4 with the (1, 4)-entry made nonzero satisfies
the surjective neighborhoods condition over F.
Proof. The following is a nilpotent realization of the superpattern with the (1, 4)
entry made nonzero when (x1 , x2 , x3 , x4 ) = −1, 2, 1, 51 .


 x1 1 x2 −1 


 4

−
1
1
0
 5





 0

0
0
x
3




x4 − 13 0 0
Set a = −1, 2, 1, 51 , and the polynomial map F : F4 → F4 maps (x1 , x2 , x3 , x4 ) to
x3
4
x1 x3 4x2 x3
4
− x4 + , −
−
+ x2 x 3 x4 − x3 x 4
−x1 − 1, x1 + x4 + , −x2 x3 x4 +
5
3
15
3
15
Setting F (x1 , x2 , x3 , x4 ) = (c1 , c2 , c3 , c4 ) ∈ F4 and solving for each xi , we get the following
system of equations:
x1 = −c1 − 1
1
5
2
2
15c1 + 30c1 c2 + 15c1 c3 + 15c1 c4 − 2c1 + 15c2 − 2c2 + 15c2 c3 − c3 + 15c2 c4 + 3c4 +
x3 =
1
−10c21 − 25c1 c2 + 5c1 − 15c22 + 4c2 + 15
x3 (10c1 + 15c2 − 2) + 15c4
x2 =
x3 (15c1 + 15c2 − 1)
x 4 = c1 + c2 +
1
15
For each i we can fix |ci | < for sufficiently small so that each xi will be nonzero and
the denominators of x2 and x3 will be nonzero. Set N = {(n1 , n2 , n3 , n4 ) ∈ F4 : |ni | < },
51
and let M be the preimage of N under F . Since F is a continuous map, M is an open
neighborhood of a in F4 , and for all (m1 , m2 , m3 , m4 ) ∈ M , mi 6= 0. Thus, the pattern
satisfies the surjective neighborhoods condition over F.
Claim 4.9 The fill 1 superpattern of M4 with the (4, 3)-entry made nonzero satisfies
the surjective neighborhoods condition over F.
Proof. The following is a nilpotent realization of the superpattern with the (4, 3)entry made nonzero when (x1 , x2 , x3 , x4 ) = (−2, −5, −1, 1)


 x1 1


 x2 2


 0 0


x3 x4
1 0 


2 0 


0 1 


1 0
Set a = (−2, −5, −1, 1) and the polynomial map F : F4 → F4 is given by:
(x1 , x2 , x3 , x4 ) 7−→ (−x1 − 2, 2x1 − x2 − 1, x1 − x3 − 2x4 + 2, 2x4 x1 − 2x1 + x2 − x2 x4 )
Setting F (x1 , x2 , x3 , x4 ) = (c1 , c2 , c3 , c4 ) ∈ F4 and solving for each xi , we get the
following system of equations:
x1 = −c1 − 2
x2 = −2c1 − c2 − 5
x3 = −
(c2 + 1) (c1 + c3 ) + 2c4 + 2c2 + 2
c2 + 1
c2 + c4 + 1
x4 =
c2 + 1
52
For values of ci sufficiently close to 0, we have each xi 6= 0. Thus, there exists
neighborhoods M and N of a and 0 respectively such that
• F : M → N is onto.
• For all (m1 , m2 , m3 , m4 ) ∈ M we have mi 6= 0.
• If N ⊂ F4 , then M ⊂ F4 .
Thus, this pattern satisfies the surjective neighborhoods conditions over F.
It is worth noting that the Jacobians of each of the above three polynomial maps F
evaluated at a are invertible, so each of these fill 1 superpatterns satisfy the nilpotentJacobian method. Hence, all superpatterns of M4 are spectrally arbitrary over R and
C.
4.2.4
Fill 2 Superpatterns of M4
Every fill 2 superpattern of M4 is permutationally similar to one of the following eight
patterns. The following is a list of all eight patterns along with their permutationally
similar patterns.
Claim 4.10 All fill 2 superpatterns of M4 are spectrally arbitrary over F.
53
1.

 ∗ ∗ ∗


 ∗ ∗ ∗


 0 0 ∗


∗ ∗ 0




∗ 


0 



∗ 


0
∗ ∗ ∗ 0 


∗ ∗ ∗ 0 



∗ 0 0 ∗ 


∗ ∗ 0 ∗
∗ ∗ ∗ 0 


∗ ∗ ∗ ∗ 



0 0 ∗ ∗ 


∗ ∗ 0 0
∗ ∗ ∗ 0 


∗ ∗ ∗ 0 


0 ∗ 0 ∗ 


∗ ∗ 0 ∗
The following is a nilpotent realization of the superpattern when (x1 , x2 , x3 , x4 ) =
7
,
17,
.
4, − 130
7
4


x2 x3 x4 
 x1


 103

 − 7 −5 −6 0 




 0

0
1
1




7
17
0
0
4
The polynomial map determined by the above matrix has nonsingular Jacobian
7
evaluated at 4, − 130
,
17,
, so by theorem 2.7 this pattern and all of its super7
4
patterns are spectrally arbitrary over F.
2.

 ∗ ∗ ∗


 ∗ ∗ ∗


 ∗ 0 ∗


∗ ∗ 0




0 


0 



∗ 


0
∗ ∗ ∗ ∗ 


∗ ∗ ∗ 0 



0 0 0 ∗ 


∗ ∗ 0 ∗
∗ ∗ ∗ 0 


∗ ∗ ∗ 0 



0 ∗ ∗ ∗ 


∗ ∗ 0 0
∗ ∗ ∗ 0 


∗ ∗ ∗ ∗ 


0 0 0 ∗ 


∗ ∗ 0 ∗
The following is a nilpotent realization of the superpattern when (x1 , x2 , x3 , x4 ) =
54
4, −22, 1, 21 .


1 0 
 x1 1




 x2 −5 −6 0 




 x3 0

1
1




x4 14
0 0
The polynomial map determined by the above matrix has nonsingular Jacobian
evaluated at 4, −22, 1, 12 , so by theorem 2.7 this pattern and all of its superpatterns are spectrally arbitrary over F.
3.

 ∗ ∗ ∗


 ∗ ∗ ∗


 0 0 ∗


∗ ∗ ∗


0 


0 



∗ 


0
∗ ∗ ∗ 0 


∗ ∗ ∗ 0 


0 0 0 ∗ 


∗ ∗ ∗ ∗
The following is a nilpotent realization of the superpattern when (x1 , x2 , x3 , x4 ) =
(1, −6, 1, 1).

1 x1
 4


 −22 −5 x2


 0
0 x3


9
5
x4
2
4

0 


0 


1 


0
The polynomial map determined by the above matrix has nonsingular Jacobian
evaluated at (1, −6, 1, 1), so by theorem 2.7 this pattern and all of its superpatterns
are spectrally arbitrary over F.
55
4.


 ∗ ∗ ∗


 ∗ ∗ ∗


 0 0 ∗


∗ ∗ 0
0 


0 


∗ 


∗
The following realization is nilpotent at (x1 , x2 , x3 , x4 ) = 3, −18, −3, 61 .


1 0 
 x1 1




 x2 −5 −6 0 





 0 0
1
1




x3 x4 0 1
Setting the polynomial map F = (f1 , f2 , f3 , f4 ) determined by the above matrix
equal to (c1 , c2 , c3 , c4 ) ∈ F4 and solving for x1 , x2 , x3 , x4 , we get the following system
of equations.
x 1 = 3 − c1
x2 = −3x1 − 9 − c2
x3 = −3c1 − 2c2 + 6x4 − 4 − c3
x4 =
c1 + c2 + c3 + c4 + 1
3c1 + c2 + 6
For each i we can fix |ci | < for sufficiently small so that each xi will be nonzero
and the denominator of x4 will be nonzero. Set N = {(n1 , n2 , n3 , n4 ) ∈ F4 : |ni | <
}, and let M be the preimage of N under F . Since F is a continuous map, M is
an open neighborhood of a in F4 , and for all (m1 , m2 , m3 , m4 ) ∈ M , mi 6= 0. Thus,
the pattern satisfies the surjective neighborhoods condition over F.
56
5.

 ∗ ∗ ∗


 ∗ ∗ ∗


 0 0 0


∗ ∗ 0


∗ 


∗ 



∗ 


0
∗ ∗ ∗ 0 


∗ ∗ ∗ 0 


∗ ∗ 0 ∗ 


∗ ∗ 0 0
The following is a nilpotent realization of the
− 54 , 1, 1, 12 .

20
15
 −1 21 7


 x1 x2 x3


 0
0 0


1
− 13 0
5
superpattern when (x1 , x2 , x3 , x4 ) =
5
− 14
x4
1
0










The polynomial map determined by the above matrix has nonsingular Jacobian
evaluated at − 54 , 1, 1, 21 , so by theorem 2.7 this pattern and all of its superpatterns
are spectrally arbitrary over F.
6.

 ∗ ∗ ∗


 ∗ ∗ ∗


 ∗ 0 0


∗ ∗ 0


∗ 


0 



∗ 


0
∗ ∗ ∗ 0 


∗ ∗ ∗ ∗ 


0 ∗ 0 ∗ 


∗ ∗ 0 0
The following is a nilpotent realization of the superpattern when (x1 , x2 , x3 , x4 ) =
57
−1, − 54 , 1, 15 .

 x1


 x2


 x3


x4
− 16
41
42
− 41
− 59
41
1
1
0
0
0
1
− 31
0
0










The polynomial map determined by the above matrix has nonsingular Jacobian
evaluated at −1, − 54 , 1, 15 , so by theorem 2.7 this pattern and all of its superpatterns are spectrally arbitrary over F.
7.

 ∗ ∗ ∗


 ∗ ∗ ∗


 0 ∗ 0


∗ ∗ 0


∗ 


0 



∗ 


0
∗ ∗ ∗ 0 


∗ ∗ ∗ ∗ 


∗ 0 0 ∗ 


∗ ∗ 0 0
The following is a nilpotent realization of
100 130
−1, 225
,
,
.
77 77 77

 x1 x2

 4
 −5 1


 0
1


1
− 13
5
the superpattern when (x1 , x2 , x3 , x4 ) =

x3 x4 


1 0 


0 1 


0 0
The polynomial map determined by the above matrix has nonsingular Jacobian
100 130
evaluated at −1, 225
,
,
, so by theorem 2.7 this pattern and all of its super77 77 77
patterns are spectrally arbitrary over F.
58
8.

 ∗ ∗ ∗


 ∗ ∗ ∗


 0 0 0


∗ ∗ ∗




∗ 


0 



∗ 


0
∗ ∗ ∗ 0 


∗ ∗ ∗ ∗ 



0 0 0 ∗ 


∗ ∗ ∗ 0
∗ ∗ ∗ 0 


∗ ∗ ∗ 0 



∗ 0 0 ∗ 


∗ ∗ ∗ 0
∗ ∗ ∗ 0 


∗ ∗ ∗ 0 


0 ∗ 0 ∗ 


∗ ∗ ∗ 0
The following is a nilpotent realization of
45 145
.
, 41 , − 230
−1, 41
41

 x1 x2

 4
 −5 1


 0
0


1
− 13
5
the superpattern when (x1 , x2 , x3 , x4 ) =

x3 x4 


1 0 


0 1 


1 0
The polynomial map determined by the above matrix has nonsingular Jacobian
145
230
evaluated at −1, 45
,
,
−
, so by theorem 2.7 this pattern and all of its su41 41
41
perpatterns are spectrally arbitrary over F.
Claim 4.11 Every superpattern of M4 is spectrally arbitrary over F.
To show that every superpattern of M4 is spectrally arbitrary over F, it suffices to
show that every superpattern of pattern 4 in the list of fill 2 superpatterns above is
spectrally arbitrary over F. Pattern 4 has two fill 1 superpatterns mod permutation
similarity, shown below.

 ∗ ∗ ∗


 ∗ ∗ ∗


 0 0 ∗


∗ ∗ 0

∗ 


0 


∗ 


∗

 ∗ ∗ ∗


 ∗ ∗ ∗


 0 0 ∗


∗ ∗ ∗

0 


0 


∗ 


∗
59
The following are a nilpotent realizations when (x1 , x2 , x3 , x4 ) = (3, 1, 1, 1) and
(y1 , y2 , y3 , y4 ) = (1, −6, 1, 1), respectively.


 x1 x2 x3 x4 



 81
 − 7 −5 −6 0 





 0
0
1
1




7
− 45
−
0
1
7
3

1 y1
 3


 −19 −5 y2


 0
0 y3


8
− 87
y4
7

0 


0 


1 


1
Both polynomial maps determined by the above matrices have nonsingular Jacobians at
(x1 , x2 , x3 , x4 ) = (3, 1, 1, 1) and (y1 , y2 , y3 , y4 ) = (1, −6, 1, 1), respectively, so by theorem
2.7 both patterns and all of their superpatterns are spectrally arbitrary over F.
60
Chapter 5
Field Extensions, The 2n Conjecture, The
Superpattern Conjecture
5.1
Spectrally Arbitrary Patterns and Field Extensions
If a pattern A is spectrally arbitrary over a field F, what (if anything) can be said about
A over a field extension E of F? Is A spectrally arbitrary over E; potentially nilpotent
over E; relaxed spectrally arbitrary over E? Does it matter if E or F is topologically
closed or algebraically closed? We can ask the same questions about A over a subfield
of F as well. The main conjecture about spectrally arbitrary patterns and subfields is
false. That is, if K ⊆ F is an extension of fields and A is spectrally arbitrary over F, we
cannot assume A is spectrally arbitrary over K (consider M4 and N4 in section 4.5).
It might be true that if a pattern is spectrally arbitrary over F where Q ⊆ F ⊆ E,
then it is spectrally arbitrary over E. There are no known counterexamples to this
conjecture.
Conjecture 5.1 If Q ⊆ F ⊆ E is an extension of fields and a pattern is spectrally
arbitrary over F, then it is spectrally arbitrary over E.
If we add the constraints that E is topologically closed and F is dense in E, we can
61
prove the following weaker version of conjecture 5.1.
Theorem 5.2 Let E be a topologically closed field with a defined metric. If A is an
n × n pattern that is spectrally arbitrary over F, where F is a dense subfield of E, then
A is relaxed spectrally arbitrary over E.
Proof. Assume A is a spectrally arbitrary n × n pattern over F with m nonzero
entries. Let F = (f1 , f2 , . . . , fn ) : Em → En be the polynomial map of A .
Fix c ∈ En . For positive real number , let B (c) denote the -neighborhood of c
(under the Euclidean norm induced by the metric) in En , and let B (c) denote the topo
n
−1
1
logical closure of B (c) in E . For each k ∈ N, set Bk = F
B (c) . By assumption
k
B 1 (c) ∩ Fn 6= ∅ for each k, so there is some strictly nonzero b = (b1 , . . . , bm ) ∈ Fm and
k
F (b) ∈ B 1 (c)∩Fn . Hence, B1 ⊇ B2 ⊇ B3 ⊇ . . . is a nested sequence of closed, nonempty
k
T
T
m
sets in E , a topologically complete space, so Bk 6= ∅. For a = (a1 , . . . , am ) ∈ Bk ,
F (a) = c. Thus, A is a relaxed spectrally arbitrary pattern over E.
Since R and C are topologically closed, we have the following two corollaries.
Corollary 5.3 Every spectrally arbitrary pattern over Q is relaxed spectrally arbitrary
over R.
Corollary 5.4 Every spectrally arbitrary pattern over Q is relaxed spectrally arbitrary
over C.
We could strengthen the theorem by asserting that A is spectrally arbitrary over R
T
if in the proof we could show that there exists (a1 , . . . , an ) ∈ Bk such that each ai 6= 0,
but attempts at this argument have so far been unsuccessful. All known patterns that
62
are spectrally arbitrary over F for Q ⊆ F ⊆ R are also spectrally arbitrary over R, but
the following conjecture remains open.
Conjecture 5.5 If A is a pattern is spectrally arbitrary over F, where F is a dense
subfield of R (C), then A is spectrally arbitrary over R (C).
There are examples of patterns that are spectrally arbitrary over C that are not
spectrally arbitrary over R (see Chapter 4), so it is not true that the property of being
spectrally arbitrary is preserved over subfields. We showed in chapter 4 that M4 is not
spectrally arbitrary over R, because R is not algebraically closed. We now show that
if F is algebraically closed and a pattern is spectrally arbitrary over a field extension
of F, then the pattern is spectrally arbitrary over F. Hence every complex spectrally
arbitrary pattern is spectrally arbitrary over Q.
Theorem 5.6 Let F ⊆ E be an extension of fields where F is algebraically closed. If A
is spectrally arbitrary over E, then A is spectrally arbitrary over F.
Proof. We assume A is an n×n pattern with m nonzero entries. Let F = (f1 , . . . , fn )
be the polynomial map of A . Fix (b1 , . . . , bn ) ∈ Fn and set I = hf1 − b1 , . . . , fn − bn i in
F[x1 , . . . , xn ]. Suppose VF (I) = ∅. By Hilbert’s (weak) Nullstellensatz, 1 ∈ I, so there
exists k1 , . . . , kn ∈ F[x1 , . . . , xm ] such that 1 = k1 (f1 − b1 ) + · · · + kn (fn − bn ). Since A is
spectrally arbitrary over E, there exists a = (a1 , . . . , am ) ∈ Em such that aj 6= 0 for all
j and fi (a) = bi for each i, implying 1 = k1 (a)(f1 (a) − b1 ) + · · · + kn (a)(fn (a) − bn ) = 0.
Hence, VF (I) 6= ∅, so A is relaxed spectrally arbitrary over F.
We now show that A is spectrally arbitrary over F by showing that there exists some
c = (c1 , . . . , cm ) ∈ VF (I) such that ci 6= 0 for all i. If this is not true, then x1 x2 · · · xm ∈
63
IF (VF (I)). By Hilbert’s (strong) Nullstellensatz, x1 x2 · · · xm ∈
√
I. Thus, there is some
positive integer d such that (x1 x2 · · · xm )d ∈ I, so there exists k1 , . . . , kn ∈ F[x1 , . . . , xm ]
such that (x1 x2 · · · xm )d = k1 (f1 − b1 ) + · · · + kn (fn − bn ). However, this leads to a contradiction, because 0 6= (a1 a2 · · · am )d = k1 (a)(f1 (a) − b1 ) + · · · + kn (a)(fn (a) − bn ) = 0.
Therefore, A is spectrally arbitrary over F.
Corollary 5.7 Every spectrally arbitrary pattern over C is spectrally arbitrary over Q.
Note that the proof of theorem 5.6 applies to any set of multivariate polynomials, so
we can generalize theorem 5.6 in the following corollary.
Corollary 5.8 Let F ⊆ E be an extension of fields where F is algebraically closed. If
a set of polynomials {g1 , . . . , gk } ⊂ F[x1 , . . . , xm ] has a common zero in Em , then they
have a common zero in Fm . In addition, if the set of polynomials has a common zero
(a1 , . . . , am ) ∈ Em such that each ai 6= 0, then there is a common zero (c1 , . . . , cm ) ∈ Fm
such that each ci 6= 0.
Informally, theorem 5.6 and its corollaries state that if a system of multivariate
polynomial with coefficients from F ⊆ Q has a common zero, then there is a common
zero whose components are all algebraic numbers. That is, transcendental numbers are
not needed to solve a system of multivariate polynomial equations where each coefficient
and all constants are algebraic numbers.
64
5.2
The Superpattern Conjecture
Recall that it has been proven ([7]) that if the nilpotent-Jacobian method is used to show
that a pattern is spectrally arbitrary over R (or C), then every one of its superpatterns
is also spectrally arbitrary. This lead to the Superpattern Conjecture: if a pattern is
spectrally arbitrary over R, then all of its superpatterns are spectrally arbitrary over
F. This conjecture was originally posed in [7] for sign patterns over R, but it has since
been generalized for zero-nonzero patterns over a general field F. There are no known
counterexamples to this conjecture over any field extension of Q, however a number of
spectrally arbitrary patterns over the finite field F3 , which have superpatterns that are
not spectrally arbitrary, are presented in this chapter.
Conjecture 5.9 (The Superpattern Conjecture) If a pattern is spectrally arbitrary
over a field F, then all of its superpatterns are spectrally arbitrary over F.
5.2.1
4 × 4 Counterexamples to the Superpattern Conjecture
over F3
Sage and Mathematica were used to establish that the pattern T (shown below) is the
smallest spectrally arbitrary 4 × 4 pattern over F3 . See appendices B and C for sample
Mathematica and Sage code, respectively.

 ∗ ∗ 0


 0 ∗ ∗
T =

 0 0 ∗


∗ 0 ∗

∗ 


∗ 


∗ 


∗
65
Observation 5.10 T is the smallest (in terms of nonzero entries) spectrally arbitrary
4 × 4 pattern over F3 .
Moreover, the fill 1 superpattern of T with the (2,1) entry made nonzero is not
spectrally arbitrary as p(t) = t4 + t2 + 2 is not realizable by a matrix with this zerononzero pattern.
Observation 5.11 The Superpattern Conjecture is false for zero-nonzero patterns over
F3 .
The fill 1 superpattern with the (3,1) entry made nonzero is also not spectrally
arbitrary as p(t) = t4 + 2 is not realized, but the fill 2 superpattern with both the
(2,1) and (3,1) entries made nonzero is spectrally arbitrary. A detailed analysis of
these patterns revealed that since w2 = 1 for nonzero w ∈ F3 adding more variables
decreased the flexibility in the coefficient functions of the the characteristic polynomial.
In analyzing all 4 × 4 patterns, more counterexamples to the superpattern conjecture
over F3 were discovered.
The following pattern realizes 81 polynomials

 ∗ ∗ ∗


 ∗ ∗ ∗
E =

 ∗ 0 0


∗ ∗ 0
over F3 , so it is spectrally arbitrary:

∗ 


∗ 


∗ 


0
All fill 1 and 2 superpatterns of E are spectrally arbitrary, but the fill 3 superpattern E 0
66
is not spectrally arbitrary, as it only realizes 79 polynomials.


 ∗ ∗ ∗ ∗ 




∗
∗
∗
∗


0

E =


 ∗ ∗ ∗ ∗ 




∗ ∗ ∗ 0
The pattern E 0 does not realize the polynomials t4 + 1 and t4 + 2t2 + 2. The full 4 × 4
pattern is also not spectrally arbitrary over F3 , as it does not realize the polynomials
t4 + t3 + 2 and t4 + 2t2 + 2.
5.2.2
5×5 and 6×6 Counterexamples to the Superpattern Conjecture
Naturally, the question as to whether larger order counterexamples exist was considered.
Perhaps this property only occurs for 4 × 4 patterns. The following examples show that
the Super Pattern Conjecture still fails for larger matrices.
The pattern C (shown below) is a 5 × 5 spectrally arbitrary pattern over F3 .

∗ ∗ 0 ∗ ∗


 0 ∗ ∗



C = ∗ 0 ∗


 0 0 0


∗ 0 0



∗ 0 



∗ 0 


∗ ∗ 


∗ 0
The fill 1 superpattern of C with the (2, 5) entry made nonzero is not spectrally
arbitrary over F3 as p(t) = t5 + 2t is not realized by this superpattern.
67
The fill 1 superpattern with the (5, 3) entry made nonzero is not spectrally arbitrary
as none of the following polynomials are realized: t5 + t2 + 2, t5 + 2t2 + 1, t5 + t4 + 2t3 +
t2 + t + 2, t5 + 2t4 + 2t3 + 2t2 + t + 1. However, the fill 2 superpattern with both the
(2,5) and (5,3) entries made nonzero is spectrally arbitrary. As with the 4 × 4 pattern
E , C is a spectrally arbitrary pattern with a fill 2 superpattern that is also spectrally
arbitrary, yet both intermediate fill 1 superpatterns are not spectrally arbitrary. Since
two such patterns have been discovered, can a family of patterns be discovered that
possess this pathological property? The question remains open, but following example
was discovered in the 6 × 6 case.
The pattern D realizes 729 polynomials, so it

 ∗ ∗ 0 ∗


 0 ∗ ∗ ∗


 ∗ 0 ∗ ∗

D =

 0 0 0 ∗


 0 0 0 ∗


∗ 0 0 ∗
is spectrally arbitrary over F3 .

0 ∗ 


0 0 


0 0 



∗ 0 


0 ∗ 


∗ 0
The fill 1 superpattern with the (4, 1) entry made nonzero is not spectrally arbitrary
as none of the following polynomials are realized: t6 + t5 + t4 + 2t + 2, t6 + t5 + t4 + t2 +
t + 1, t6 + 2t5 + t4 + t + 2, t6 + 2t5 + t4 + t2 + 2t + 1.
The fill 1 superpattern with the (4, 2) entry made nonzero is not spectrally arbitrary
as none of the following polynomials are realized: t6 + 2t4 + 2, t6 + 2t4 + 2t2 + 2, t6 + t5 +
2t4 +2t3 +2, t6 +t5 +2t4 +2t3 +2t2 +t+1, t6 +2t5 +2t4 +t3 +2, t6 +2t5 +2t4 +t3 +2t2 +2t+1.
The fill 1 superpattern with the (4, 3) entry made nonzero is not spectrally arbitrary
as t6 + t4 + 2 and t2 + 2 are not realized, and the fill 1 superpattern with the (4, 6) entry
68
made nonzero is not spectrally arbitrary as t6 + 2t4 + 2t2 + 2 is not realized. All other
fill 1 superpatterns are spectrally arbitrary over F3 .
All fill 2 superpatterns of D with any two of the entries (4, 1), (4, 2), (4, 3), (4, 6)
made nonzero are spectrally arbitrary. Thus, D is another example of a spectrally arbitrary pattern with fill 2 superpatterns that are spectrally arbitrary, yet some intermediate
fill 1 superpatterns are not spectrally arbitrary.
It is the author’s contention that higher order counterexamples to the superpattern
conjecture over F3 can be found. Other methods need to be developed to discover
such examples, as direct computation using Sage and Mathematica becomes somewhat
unfeasible in patterns of order 7 and greater.
It is unclear whether these examples can be generalized to fields of higher cardinality,
or if they are a result of the pathological nature of F3 . These examples show that the
superpattern conjecture is not true over a general field F, but it may still hold over R,
C, and extensions of Q.
69
Conjecture 5.12 If a pattern is spectrally arbitrary over F, where Q ⊆ F ⊆ C, then all
of its superpatterns are spectrally arbitrary over F.
5.3
Lower Bound of Nonzero Entries in Spectrally
Arbitrary Patterns
In[1], Britz, McDonald, Olesky and van den Driesshe showed that an irreducible n × n
(sign) pattern must contain at least 2n − 1 nonzero entries in order to be spectrally
arbitrary over R (and C). It is believed that this bound is too liberal and it can be
strengthened to 2n for all irreducible zero-nonzero patterns. The authors in [8] showed
2n is the correct bound for n = 3 and n = 5 over R, and the authors in [6] showed 2n
is the correct bound for n = 4 over R. This is known as the 2n Conjecture, and every
known irreducible spectrally arbitrary pattern over R (and C) satisfies this bound.
Conjecture 5.13 (The 2n Conjecture) Let A is an irreducible n × n pattern with
m nonzero entries. If A is spectrally arbitrary over a field F, then m ≥ 2n.
Note that this bound on nonzero entries on irreducible spectrally arbitrary patterns
is sharp. The n × n companion-style pattern Cn has 2n nonzero entries and we showed
in chapter 2 that it is spectrally arbitrary over every field extension of Q.
5.3.1
The 2n Conjecture and Finite Fields
We also consider the lower bound of nonzero entries necessary in order for a pattern
to be spectrally arbitrary over a finite field. At an AIM workshop[26], Bryan Shader
proved that the 2n Conjecture holds over finite fields:
70
Theorem 5.14 (Shader) For n ≥ 2, if A is an irreducible n×n pattern with k nonzero
entries that is spectrally arbitrary over a field Fq , then (q − 1)k−(n−1) ≥ q n . In particular,
k ≥ 2n.
Proof. By Lemma 1.1, it is safe to assume there are k − (n − 1) nonzero entries left
as variables in A , so there are at most (q − 1)k−(n−1) non-equivalent realizations of A
over Fq . Since there are q n possible monic polynomials of degree n over Fq and A is
spectrally arbitrary over Fq , (q − 1)k−(n−1) ≥ q n must be satisfied. Since (q − 1) < q it
must be true that k − (n − 1) > n, which implies k ≥ 2n.
If A is an n × n irreducible spectrally arbitrary pattern over a finite field Fq with
k nonzero entries, then the inequality (q − 1)k−(n−1) ≥ q n must be satisfied. Thus,
log(q)
k ≥ 1 + log(q−1)
n − 1, and given a field Fq and fixed n it is natural to ask how close
one can get to this bound. In particular, what is the smallest (in terms of nonzero
entries) pattern that is spectrally arbitrary over F3 ? In [18], Bodine proved that no 3 × 3
pattern is spectrally arbitrary over F3 .
Also, note that the smallest 4 × 4 spectrally arbitrary pattern over F3 has 11 nonzero
entries, the smallest (found) 5 × 5 spectrally arbitrary pattern has 14 nonzero entries,
and the smallest (found) 6 × 6 spectrally arbitrary pattern has 17 nonzero entries (see
log(q)
section 5.2.2). The bound given by Shader[26], k ≥ 1 + log(q−1)
n − 1, yields lower
bounds of 10, 12, and 15 for order 4, 5, and 6 SAPs over F3 , respectively. It would be
interesting to know if this pattern continues or if there is an n × n spectrally arbitrary
l
m
log(3)
pattern over F3 that has
1 + log(2)
n − 1 nonzero entries, where dze denotes the
smallest integer greater than z.
71
5.3.2
Lower Bound of Nonzero Entries in Spectrally Arbitrary
Patterns over Extensions of Q
In [1] the authors used the notion of transcendental degree in proving that every every
real spectrally arbitrary pattern has at least 2n − 1 nonzero entries. We will utilize
their argument along with some results from section 5.1 to prove that if a pattern A is
spectrally arbitrary over F where F is a dense subfield of R or C, then A has at least
2n − 1 nonzero entries. The following lemma is needed to prove this result. This lemma
is a corollary of lemma 1.4, and it was given in a paper by Cavers and Fallat (see section
4 in [13]). The terminology has been changed to fit the definitions and notation of this
dissertation.
Lemma 5.15 (Cavers and Fallat) Let B be an n×n irreducible zero-nonzero pattern
and let B 0 be the relaxation of B. For every realization B of B 0 , there is a diagonal matrix D such that D−1 BD has at least n − 1 off-diagonal entries in {0, 1} that correspond
to ∗-entries in B.
We now define the notion of algebraic independence and transcendental degree. A
set S ⊆ R (or S ⊆ C) is algebraically independent if, for all s1 , s2 , . . . , sn ∈ S, if
p(s1 , s2 , . . . , sn ) = 0 for p(x1 , x2 , . . . , xn ) ∈ Q[x1 , x2 , . . . , xn ], then p is the zero polynomial. Let Q(S) denote the field of rational expressions
Q(S) =
p(s1 , . . . , sm )
: p, q ∈ Q[x1 , . . . , xn ], s1 , . . . , sn ∈ S
q(sm+1 , . . . , sn )
We define a transcendental base for Q(S) over Q to be the maximal subset of S (with
respect to inclusion) that is algebraically independent over Q. As transcendental bases
are size invariant (see [9] for further details), the transcendental degree of Q(S) over
72
Q, denoted tr.d.Q(S), is the cardinality of a transcendental base. We are now ready to
prove the result.
Theorem 5.16 Every irreducible n × n pattern that is spectrally arbitrary over F where
F is a dense subfield of R or C has at least 2n − 1 nonzero entries.
Proof. Assume A is a spectrally arbitrary zero-nonzero pattern over F with m
nonzero entries, and let A 0 be the relaxation of A . Let {α1 , . . . , αn } be an algebraically
independent set in R (or C). By theorem 5.2 there is a realization B = (bij ) of A 0 such
that the characteristic polynomial of B equals tn + α1 tn−1 + · · · + αn . By lemma 5.15,
we can assume B has at least n − 1 entries in the set {0, 1} that correspond to ∗-entries
in A . Since each αi is a polynomial in the entries {bij : 1 ≤ i, j ≤ n}, Q(α1 , . . . , αn ) ⊆
Q(bij : 1 ≤ i, j ≤ n). Thus, tr.d.Q(α1 , . . . , αn ) ≤ tr.d.Q(bij : 1 ≤ i, j ≤ n).
Since n − 1 of the entries of B that correspond to ∗-entries in A are 0 or 1, we have
tr.d.Q(bij : 1 ≤ i, j ≤ n) ≤ m − (n − 1). Hence, n = tr.d.Q(α1 , . . . , αn ) ≤ tr.d.Q(bij :
1 ≤ i, j ≤ n) ≤ m − (n − 1), implying m ≥ 2n − 1.
Corollary 5.17 Every irreducible n × n spectrally arbitrary pattern over Q or Q has at
least 2n − 1 nonzero entries.
It is worth noting that there numerous examples of relaxed spectrally arbitrary patterns with exactly 2n − 1 nonzero entries such as the relaxed companion pattern shown
below, but every known example is also not spectrally arbitrary as zeros are needed to
satisfy particular monic polynomials. Consider the following example.
73
Example 5.18 The n × n relaxed companion pattern is relaxed spectrally arbitrary over
all fields F, but each ]-entry in the first column needs to be zero in order to realize the
characteristic polynomial tn .







Cn0 = 





] ]
]
]
..
.
]
0
·
0



. 0 
0 ]



.. ..
.
. 0 
0


.. . .
. 0 ] 
.


0 ··· 0 0
..
74
Chapter 6
Concluding Remarks
In this dissertation, we explored the problem of classifying zero-nonzero patterns that
are spectrally arbitrary over field extensions of Q. We developed a sufficient technique,
called the surjective neighborhoods condition, for determining if zero-nonzero patterns
are spectrally arbitrary over field extensions of Q by closely analyzing the proof of the
nilpotent-Jacobian method. We then showed that in certain cases the nilpotent-Jacobian
method can be used to prove that a pattern and all of its superpatterns are spectrally
arbitrary over field extensions of Q. We utilized these results on each real and complex
3 × 3 and 4 × 4 spectrally arbitrary pattern to determine over which field extensions
of Q that it is spectrally arbitrary. In doing this, we came up with several interesting
conjectures and observations, which we summarize below.
1. If a pattern is real spectrally arbitrary, is it spectrally arbitrary over the real
algebraic closure of Q?
2. If a pattern is complex spectrally arbitrary, is it spectrally arbitrary over Q?
3. If a pattern is rational spectrally arbitrary, is it real spectrally arbitrary?
4. If a pattern is spectrally arbitrary over some field extension of Q, are every one of
its superpatterns spectrally arbitrary over that field?
75
We can answer each these questions affirmatively if we restrict ourselves to 3 × 3 and
4 × 4 patterns. We proved question 2 affirmatively for all n × n patterns and we proved
a weaker version of question 3 for n × n patterns in chapter 5. It is the hope of the
author that all of these questions will eventually be answered for n × n patterns by utilizing techniques developed in this work, techniques not yet developed, and results from
various branches of mathematics including algebraic geometry, differentiable geometry,
field theory, commutative algebra, topology, and combinatorics.
In chapter 5 we provided a number of counterexamples to the superpattern conjecture
over F3 . It is the author’s contention that a family of n × n counterexamples to the
superpattern conjecture over F3 exists, but a more efficient algorithm must be developed
to determine if large order patterns realize each possible characteristic polynomial over
F3 .
Finally, in chapter 5 we proved that the number of nonzero entries of any spectrally
arbitrary pattern over any field extension of Q must be at least 2n − 1. This bound can
likely be strengthened to 2n, but the 2n conjecture remains an open problem.
The study of zero-nonzero patterns over arbitrary fields and field extensions of Q is
a new area of research in combinatorial matrix theory that can yield important results
linking the combinatorial structure of a pattern with certain properties of a field in determining whether the pattern is spectrally arbitrary over the field. The nilpotent-Jacobian
method is a powerful tool in analyzing patterns over R and C, but other techniques and
avenues of research need to be developed to analyze patterns over other fields. It is our
hope that these open conjectures and unanswered questions will eventually be solved,
and that some of the techniques developed in this paper can be utilized in answering
these questions.
76
Appendix A
4 × 4 Minimally Spectrally Arbitrary Patterns Over
R or C
In this appendix let F denote an arbitrary field extension of Q.
A.1
4 × 4 Real Minimally Spectrally Patterns with 8
Nonzero Entries
The first 9 patterns are spectrally arbitrary over every field extension F of Q, including
Q. The tenth pattern is spectrally arbitrary over Q√+ , but it is not spectrally arbitrary
over any finite field extension of Q.
1. Consider the following pattern with nilpotent realization.



 ? ? 0 0 
 −1 1 0 0






 ? 0 ? ? 
 −1 0 1 1






 0 0 ? ? 
 0 0 1 1






? 0 0 0
−1 0 0 0










We change four nonzero entries to indeterminants as follows and calculate the
77
polynomial map

 x1 1


 x2 0


 0 0


x3 0
F.

0
0 


1 1 


1 x4 


0 0
F = (−x1 − 1, x1 − x2 , x2 − x3 , x3 − x3 x4 )
The Jacobian of F at a = (−1, −1, −1, 1) is invertible, so there exists neighborhoods M and N of a and (0, 0, 0, 0) ,respectively, such that F : M → N is a
bijection and for all (m1 , m2 , m3 , m4 ) ∈ M , mi 6= 0. Let q = (q1 , q2 , q3 , q4 ) ∈ F4 ∩N
and set F (x1 , x2 , x3 , x4 ) = q, so there exists a unique solution (b1 , b2 , b3 , b4 ) ∈ M .
Now b1 = −q1 − 1 ∈ F, so b2 = b1 − q2 ∈ F and hence b3 = b2 − q3 ∈ F. We see that
b4 ∈ F after solving for it in the equation b3 − b3 b4 = q4 . Hence by the surjective
neighborhoods conjecture, this pattern is spectrally arbitrary over F.
2. Consider the following pattern

 ? ? 0


 ? 0 ?


 0 0 ?


? ? 0
with nilpotent realization.


0 
 −1 1 0 0




0 
 −1 0 1 0




 0
0 1 1
? 




1 −1 0 0
0










We change four nonzero entries to indeterminants as follows and calculate the
polynomial map F .

 x1 1 0


 x2 0 1


 0 0 1


x3 x4 0

0 


0 


1 


0
F = (−x1 − 1, x1 − x2 , x2 − x4 , x1 x4 − x3 )
78
As with pattern 1, the Jacobian of F at a = (−1, −1, 1, −1) is invertible, so
F : M → N is a bijection for real neighborhoods M and N of a and (0, 0, 0, 0),
respectively, and M and for all (m1 , m2 , m3 , m4 ) ∈ M , mi 6= 0. We can use the
same elimination argument used in pattern 1 to show that for q = (q1 , q2 , q3 , q4 ) ∈
F4 there exists b = (b1 , b2 , b3 , b4 ) ∈ F4 ∩ M such that F (b) = q, so this pattern
satisfies the surjective neighborhoods condition over F.
3. Consider the following pattern

 ? ? 0


 ? 0 ?


 0 0 ?


? 0 ?
with nilpotent realization.


0 
 1 1 0 0



 1
0 
 −2 0 1 0





 0 0 1 1
? 



1
0
0 − 12 0
4










We change four nonzero entries to indeterminants as follows and calculate the
polynomial

 x1


 x2


 0


x3
map F .

1
0
0
1
0
1
0 x4
0 


0 


1 


0
F = (−x1 − 1, x1 − x2 − x4 , x2 + x1 x4 , x2 x4 − x3 )
The Jacobian of F at a = −1, − 21 , 14 , − 12
is invertible, so F : M → N is a
bijection for real neighborhoods M and N of a and (0, 0, 0, 0), respectively, and
for all (m1 , m2 , m3 , m4 ) ∈ M , mi 6= 0. Let q = (q1 , q2 , q3 , q4 ) ∈ F4 ∩ N , so there
exists b = (b1 , b2 , b3 , b4 ) ∈ M that solves F (b) = q. Clearly, b1 ∈ F, so the system
b1 − b2 − b4 = q2 , b2 + b1 b4 = q3 yields b2 , b4 ∈ F. Thus, b3 = b2 b4 − q4 ∈ F, which
proves that this pattern satisfies the surjective neighborhoods condition over F.
79
4. Consider the following pattern

 ? ? 0


 ? 0 ?


 0 0 0


? ? 0
with nilpotent realization.


1 0 0
0 
 1




0 
 −1 0 1 0




 0
0 0 1
? 




1 −1 0 1
?










We change four nonzero entries to indeterminants as follows and calculate the
polynomial map F .

 x1 1 0


 x2 0 1


 0 0 0


x3 x4 0

0 


0 


1 


1
F = (−x1 − 1, x1 − x2 , x2 − x4 , x1 x4 − x3 )
Since F is the same as polynomial map as in pattern 2, this pattern satisfies the
surjective neighborhoods condition over F.
5. Consider the following pattern

 ? ? 0


 ? 0 ?


 0 0 0


? 0 ?
with nilpotent realization.


0 
 −1 1 0 0



 1
0 
 −2 0 1 0




 0 0 0 1
? 




1
?
0 − 12 1
4










We change four nonzero entries to indeterminants as follows and calculate the
polynomial map F .

 x1 1 0 0


 x2 0 1 0


 0 0 0 1


x3 0 x4 1










F = (−x1 − 1, x1 − x2 − x4 , x2 + x1 x4 , x2 x4 − x3 )
80
Since F is the same as polynomial map as in pattern 3, this pattern satisfies the
surjective neighborhoods condition over F.
6. Consider the following pattern with nilpotent realization.



 −1 1 0 −1
 ? ? 0 ? 






 −1 0 1 0
 ? 0 ? 0 






 0
 0 0 ? ? 
0 1 1






0 − 21 0 0
0 ? 0 0










We change four nonzero entries to indeterminants as follows and calculate the
polynomial map F .


 x1 x2 0 x3 




 −1 0 x4 0 




 0
0 1 1 




1
0 −2 0 0
x3 x4 x3 x1 x4 F = −x1 − 1, x1 + x2 , −x1 −
+ ,
−
2
2 2
2
The Jacobian of F at a = (−1, 1, −1, 1) is invertible, so F : M → N is a bijection
for real neighborhoods M and N of a and (0, 0, 0, 0), respectively, and for all
(m1 , m2 , m3 , m4 ) ∈ M , mi 6= 0. We can use the same elimination argument used
above to show that for q = (q1 , q2 , q3 , q4 ) ∈ F4 ∩ N , there exists b = (b1 , b2 , b3 , b4 ) ∈
F4 ∩M such that F (b) = q. Thus, this pattern satisfies the surjective neighborhoods
condition over F.
81
7. Consider the following pattern

 ? ? 0


 ? 0 ?


 0 0 ?


0 ? 0
with nilpotent realization.


0 
 −1 1 0 0



 1
? 
 −2 0 1 1




 0
? 
0 1 1




0
0 − 12 0 0










We change four nonzero entries to indeterminants as follows and calculate the
polynomial map F .


 −1 1 0 0 




 x1 0 x2 x3 




 0
0 x4 1 




1
0 −2 0 0
x3
x2 x3
x3 x4 x2 x3 x4 F = 1 − x4 , −x1 +
− x4 ,
+
+ x1 x4 −
,
−
2
2
2
2
2
2
The Jacobian of F at a = − 12 , 1, 1, 1 is invertible, so F : M → N is a bijection
for real neighborhoods M and N of a and (0, 0, 0, 0), respectively, and for all
(m1 , m2 , m3 , m4 ) ∈ M , mi 6= 0. Let q = (q1 , q2 , q3 , q4 ) ∈ F4 ∩ N , so there exists
a solution b = (b1 , b2 , b3 , b4 ) ∈ M to the system F (b) = q. Note that b4 ∈ F,
so the last three equations of F are linear when x4 is replaced with b4 . Since a
(unique) solution, i.e. (b1 , b2 , b3 ), exists to this linear system, the solution will lie
in F. Thus, this pattern satisfies the surjective neighborhoods condition over F.
82
8. Consider the following pattern with nilpotent realization.



 0 ? 0 ? 
 0 −1 0 −1






1
0
 ? 0 ? 0 
 1 0






 0 0 ? ? 
 0 0 −1 1






0 ? 0 ?
0 1
0
1










We change four nonzero entries to indeterminants as follows and calculate the
polynomial map F .


 0 x1 0 x2


 1 0 1 0


 0 0 −1 x3


0 1 0 x4









F = (1 − x4 , −x1 − x4 , x1 x4 − x1 − x2 − x3 , x1 x4 − x2 )
The Jacobian of F at a = (−1, −1, 1, 1) is invertible, so F : M → N is a bijection
for some real neighborhoods M and N of a and (0, 0, 0, 0), respectively, and for all
(m1 , m2 , m3 , m4 ) ∈ M , mi 6= 0. Let q = (q1 , q2 , q3 , q4 ) ∈ F4 ∩ N , so there exists a
solution b = (b1 , b2 , b3 , b4 ) ∈ M to the system F (b) = q. We have b4 = 1 − q1 ∈ F,
so b1 = −q2 − b4 ∈ F. Hence, b2 = b1 b4 − q4 ∈ F, implying b3 = b1 b4 − b1 − q3 ∈ F.
Thus, this pattern is spectrally arbitrary over F by the surjective neighborhoods
conjecture.
83
9. Consider the following pattern with nilpotent realization.



1
2
 ? 0 0 ? 
 −1 0 0






 ? 0 ? ? 
 1 0 − 12 −1






 0 0 ? ? 
 0 0 1
1






0 ? 0 ?
0 1 0
1










We change four nonzero entries to indeterminants as follows and calculate the
polynomial map F .

 x1


 x2


 0


0
0
0
0 x3
0
1
1
0
1
2




x4 


1 


1
x2
x2
F = −x1 − 1, x1 − x4 , − − x3 + x1 x4 + x4 ,
+ x1 x3 − x1 x4
2
2
The Jacobian of F at a = (−1, −1, 1, 1) is invertible, so F : M → N is a bijection
for some real neighborhoods M and N of a and (0, 0, 0, 0), respectively, and for all
(m1 , m2 , m3 , m4 ) ∈ M , mi 6= 0. Let q = (q1 , q2 , q3 , q4 ) ∈ F4 ∩ N , so there exists a
solution b = (b1 , b2 , b3 , b4 ) ∈ M to the system F (b) = q. Considering the first two
equations in F , we get b1 , b4 ∈ F. Substituting these into x1 and x4 in the last two
equations in F yields a linear system, which gives b2 , b3 ∈ F. Hence, this pattern
is spectrally arbitrary over F by the surjective neighborhoods conjecture.
10. Consider the following pattern with nilpotent realization.
84


 ? ? 0


 ? 0 ?


 0 0 ?


0 ? ?
Since
√
0 


0 


? 


0


1
0
 −1
 √
 3+ 5
0
1
 2


 0
0
1


√
√
0
−2 − 5 − 1+2 5
0 


0 


1 


0
5 is needed in every nilpotent realization, so this pattern is not spectrally
arbitrary over Q. Consider the following

 v 1


 x 0


 0 0


0 y
realization.

0 0 


1 0 


x 1 


z 0
By setting the polynomial map F (v, w, x, y, z) determined by the above matrix
equal to (q1 , q2 , q3 , q4 ) where each qi ∈ Q√+ and solving, we get the following
system of equations.
v = −x − q1
z=
1
√
± γ + 2q1 x + q12 − q2 + x2
2
w = −q1 x − q2 − x2 − z
y = −q1 x2 − q2 x − q1 z − q3 − x3 − 2xz
Where γ = 12q1 x3 +10q12 x2 +2q2 x2 +4q13 x+4q3 x+q14 +q22 −2q12 q2 +4q1 q3 −4q4 +5x4 .
We can pick x ∈ Q√+ large enough such that v, z, w, y are nonzero and lie in Q√+ ,
so this pattern is spectrally arbitrary over Q√+ .
85
A.2
4 × 4 Real Spectrally Patterns with 9 Nonzero
Entries
The first 14 patterns are spectrally arbitrary over every field extension Q, including Q.
Patterns 15, 16, and 17 are spectrally arbitrary over Q√+ , and pattern 18 is spectrally
arbitrary over Qβ .
1. Consider the following pattern and nilpotent realization.

 ? ? 0


 ? ? ?


 0 0 ?


? 0 0

? 


0 


? 


0


 1 1 0 −8 




 5 1 −16 0 




 0 0 −2 1 




1 0 0
0
We change four nonzero entries to indeterminants as follows.


 x1 1 0 −8 




 x2 1 −16 0 




 0 0 −2 x4 




x3 0 0
0
For x = (x1 , x2 , x3 , x4 ), the polynomial map F determined by this matrix is given
by:
x 7−→ (1 − x1 , −x1 − x2 + 8x3 − 2, 2x1 − 2x2 + 8x3 , 16x3 x4 − 16x3 )
The Jacobian of F at a = (1, 5, 1, 1) is invertible, so F : M → N is a bijection
for some real neighborhoods M and N of a and (0, 0, 0, 0), respectively, and for
86
all (m1 , m2 , m3 , m4 ) ∈ M , mi 6= 0. Let q = (q1 , q2 , q3 , q4 ) ∈ F4 ∩ N , so there exists
a solution b = (b1 , b2 , b3 , b4 ) ∈ M to the system F (b) = q. Since the first three
equations of F are linear, we have b1 , b2 , b3 ∈ F, which implies b4 ∈ F. Hence, this
pattern is spectrally arbitrary over F by the surjective neighborhoods condition.
2. Consider the following pattern and nilpotent realization.



1 −1
 1
 ? ? ? 0 






 −1 −1 1
 ? ? ? ? 






 0
 0 0 0 ? 
0
0






1
0
0
? 0 0 0

0 


1 


1 


0
Changing the second row to 4 indeterminates yields a polynomial map with nonsingular determinate at (−1, −1, 1, 1). By theorem 2.7 this pattern and all of its
superpatterns are over F.
3. Consider the following pattern and nilpotent realization.



0
 2 1 4
 ∗ ∗ ∗ 0 






0
 ∗ 0 ∗ 0 
 −3 0 3






 0 0 ∗ ∗ 
 0 0 −1 1






∗ 0 0 ∗
1 0 0 −1










We change four nonzero entries to indeterminants as follows .


 x1 x2 x3 0 




 −3 0 x4 0 




 0 0 −1 1 




1 0 0 −1
87
The polynomial map F determined by this matrix is given by:
(x1 , x2 , x3 , x4 ) 7−→ (2 − x1 , −2x1 + 3x2 + 1, −x1 + 6x2 − x3 , 3x2 − x2 x4 )
The Jacobian of F at a = (2, 1, 4, 3) is invertible, so F : M → N is a bijection
for some real neighborhoods M and N of a and (0, 0, 0, 0), respectively, and for
all (m1 , m2 , m3 , m4 ) ∈ M , mi 6= 0. Let q = (q1 , q2 , q3 , q4 ) ∈ F4 ∩ N , so there exists
a solution b = (b1 , b2 , b3 , b4 ) ∈ M to the system F (b) = q. Since the first three
equations of F are linear, we have b1 , b2 , b3 ∈ F, which implies b4 ∈ F. Hence, this
pattern is spectrally arbitrary over F by the surjective neighborhoods condition.
4. Consider the following pattern

 ∗ ∗ 0


 ∗ 0 ∗


 0 ∗ ∗


∗ 0 0
and nilpotent realization.


0 
 −2 1 4



 5
0 
 − 3 0 −1




 0 4 1
∗ 


3


∗
1 0 0

0 


0 


1 


1
We change four nonzero entries to indeterminants as follows and calculate the
polynomial map F .


 x1 1 0 0 




 x2 0 −1 0 




 0 x4 1 1 




x3 0 0 1
F = (−2 − x1 , 2x1 − x2 + x4 + 1, −x4 x1 − x1 + 2x2 − x4 , −x2 + x3 + x1 x4 )
The Jacobian of F at a = −2, − 53 , 1, 34 is invertible, so F : M → N is a bijection for some real neighborhoods M and N of a and ~0, respectively, and for all
88
(m1 , m2 , m3 , m4 ) ∈ M , mi 6= 0. Let F ⊇ Q and q = (q1 , q2 , q3 , q4 ) ∈ F4 ∩ N , so
there exists a solution b = (b1 , b2 , b3 , b4 ) ∈ M to the system F (b) = q. Note that
b1 ∈ F, so the second two equations of F are linear in x2 and x4 after replacing x1
with b1 . Thus, b2 , b4 ∈ F, which gives b3 ∈ F.
5. Consider the following pattern

 ∗ ∗ 0


 ∗ 0 ∗


 0 ∗ 0


∗ 0 0
and nilpotent realization.


1
0 
 −1 − 2 0




∗ 
0 1
 1




 0 −1 0
∗ 


2


∗
1
0 0

0 


1 


1 


1
We change four nonzero entries to indeterminants as follows.


1
 x1 − 2 0 0 




 x2 0 1 x4 




1
 0 − 0 1 


2


x3 0 0 1
For x = (x1 , x2 , x3 , x4 ), the polynomial map F determined by this matrix is given
by:
x2 1 x1 x2 x 3 x4 1 x1 x3
x 7−→ −x1 − 1, x1 +
+ ,− −
+
− ,
+
2
2
2
2
2
2 2
2
The Jacobian of F at a = (−1, 1, 1, 1) is invertible, so F : M → N is a bijection
for some real neighborhoods M and N of a and (0, 0, 0, 0), respectively, and for all
(m1 , m2 , m3 , m4 ) ∈ M , mi 6= 0. Let q = (q1 , q2 , q3 , q4 ) ∈ F4 ∩ N , so there exists a
solution b = (b1 , b2 , b3 , b4 ) ∈ M to the system F (b) = q. Since b1 ∈ F, b2 , b3 ∈ F.
Thus, b4 ∈ F. Hence, this pattern is spectrally arbitrary over F by the surjective
neighborhoods condition.
89
6. Consider the following pattern

 0 ∗ ∗


 ∗ 0 ∗


 0 0 ∗


∗ ∗ 0
and nilpotent realization.



0 
 0 −1 −1 0 






0 
1 0 
 1 0






1
 0 0 −1

∗ 


2 



∗
1 1
0 1
We change four nonzero entries to indeterminants as follows and calculate the
polynomial map F .


 0 x4 −1 0


 1 0 1 0


 0 0 −1 1

2

x1 x2 0 x3









For x = (x1 , x2 , x3 , x4 ), the polynomial map F determined by this matrix is given
by:
x1 x2
x2 x1 x4
x 7−→ 1 − x3 , −x3 − x4 ,
−
+ x3 x4 − x4 ,
−
+ x3 x4
2
2
2
2
The Jacobian of F at a = (1, 1, 1, −1) is invertible, so F : M → N is a bijection
for some real neighborhoods M and N of a and (0, 0, 0, 0), respectively, and for
all (m1 , m2 , m3 , m4 ) ∈ M , mi 6= 0. Let q = (q1 , q2 , q3 , q4 ) ∈ F4 ∩ N , so there
exists a solution b = (b1 , b2 , b3 , b4 ) ∈ M to the system F (b) = q. From the first two
equations of the polynomial map, we have b3 , b4 ∈ F. The last two equations of the
polynomial map are linear in x1 , x2 after substituting b3 , b4 for x3 , x4 , respectively.
Since there is a unique solution, we know that b1 , b2 also lie in F. Hence, this
pattern is spectrally arbitrary over F by the surjective neighborhoods condition.
90
7. Consider the following pattern and nilpotent realization.



0
 0 ∗ 0 0 
 0 −1 0






1 −1
 ∗ 0 ∗ ∗ 
 1 0






 ∗ 0 ∗ ∗ 
 1 0 −1 3






∗ 0 0 ∗
1 0
0
1










We change four nonzero entries to indeterminants as follows.


 0 x1 0 0 




1
0
1
x

2 




 1 0 x3 x4 




1 0 0 1
For x = (x1 , x2 , x3 , x4 ), the polynomial map F determined by this matrix is given
by:
x 7−→ (−x3 − 1, x3 − x1 , x1 x3 − x1 x2 , x2 x3 x1 − x3 x1 − x4 x1 + x1 )
The Jacobian of F at a = (−1, −1, −1, 3) is invertible, so F : M → N is a
bijection for some real neighborhoods M and N of a and (0, 0, 0, 0), respectively,
and for all (m1 , m2 , m3 , m4 ) ∈ M , mi 6= 0. Let q = (q1 , q2 , q3 , q4 ) ∈ F4 ∩ N , so
there exists a solution b = (b1 , b2 , b3 , b4 ) ∈ M to the system F (b) = q. From
the first two equations of the polynomial map b3 , b1 ∈ F, so b2 ∈ F from the third
equation. Solving for b4 in the fourth equation of the polynomial map yields b4 ∈ F.
Thus, this pattern is spectrally arbitrary over F by the surjective neighborhoods
condition.
91
8. Consider the following pattern

 ∗ ∗ 0


 0 ∗ ∗


 ∗ 0 0


∗ ∗ 0
and nilpotent realization.


0 
 1 1 0 0




∗ 
 0 −1 1 −1




 1 0 0 1
∗ 




0
1 1 0 0










We change four nonzero entries to indeterminates as follows.


 x1 x2 0 0 




0
−1
1
x

3 




 x4 0 0 1 




1 1 0 0
For x = (x1 , x2 , x3 , x4 ), the polynomial map F determined by this matrix is given
by:
x 7−→ (1 − x1 , −x3 − x1 , x1 x3 − x2 x3 − x2 x4 − 1, x1 − x2 )
The Jacobian of F at a = (1, 1, −1, −1) is invertible, so F : M → N is a bijection
for some real neighborhoods M and N of a and (0, 0, 0, 0), respectively, and for
all (m1 , m2 , m3 , m4 ) ∈ M , mi 6= 0. Let q = (q1 , q2 , q3 , q4 ) ∈ F4 ∩ N , so there
exists a solution b = (b1 , b2 , b3 , b4 ) ∈ M to the system F (b) = q. From the first two
equations of the polynomial map b4 , b1 ∈ F, so the last two equations can be linearly
solved for b2 and b3 in terms of b1 , b2 , q1 , q2 , q3 , q4 ∈ F. Thus, b2 , b3 ∈ F, so this
pattern is spectrally arbitrary over F by the surjective neighborhoods condition.
92
9. Consider the following pattern and nilpotent realization.



 0 ∗ 0 0 
 0 1 0 0






 0 ∗ ∗ ∗ 
 0 −1 1 1





 1
 ∗ 0 0 ∗ 



 2 0 0 1



1
∗ ∗ 0 ∗
−1 0 1
2










We change four nonzero entries to indeterminants as follows and calculate the
polynomial map F .


 0 1 0 0 




 0 −1 1 x1 




 x2 0 0 x3 




1
−1
0
x
4
2
The polynomial map F (x1 , x2 , x3 , x4 ) determined by this matrix is given by:
x1
x3 (x1 , x2 , x3 , x4 ) 7−→ 1 − x4 , x1 − x4 , − − x2 + x3 , x2 x4 −
2
2
The Jacobian of F at a = 1, 21 , 1, 1 is invertible, so F : M → N is a bijection
for some real neighborhoods M and N of a and (0, 0, 0, 0), respectively, and for
all (m1 , m2 , m3 , m4 ) ∈ M , mi 6= 0. Let q = (q1 , q2 , q3 , q4 ) ∈ F4 ∩ N , so there
exists a solution b = (b1 , b2 , b3 , b4 ) ∈ M to the system F (b) = q. From the first two
equations of the polynomial map, we have b1 , b4 ∈ F. The last two equations of the
polynomial map are linear in x2 , x3 after substituting b1 , b4 for x1 , x4 , respectively.
Since (b1 , b2 , b3 , b4 ) is a unique solution to F (x) = q, we know that b2 , b3 also lie in F.
Hence, this pattern is spectrally arbitrary over F by the surjective neighborhoods
condition.
93
10. Consider the following pattern and nilpotent realization.



0
0
 ∗ ∗ 0 0 
 1 1






 ∗ ∗ ∗ ∗ 
 1 −1 −4 −4






 0 0 0 ∗ 
 0 0
0
1






0 ∗ ∗ 0
0 1
2
0










Replacing the second row with 4 indeterminants yields a nonsingular Jacobian at
(1, −1, −4, −4), so this pattern and all of its superpatterns are spectrally arbitrary
over F by theorem 2.7.
11. Consider the following pattern

 0 ∗ 0


 ∗ 0 ∗


 0 0 ∗


0 ∗ ∗
and nilpotent realization.


0
0 
 0 1 0




∗ 
 1 0 −1 −1





 0 0 1
1
∗ 



0 1 −1 −1
∗










We change four nonzero entries to indeterminants as follows.


 0 1 0 0 




 1 0 x1 x 2 




 0 0 x3 1 




0 1 x4 −1
For x = (x1 , x2 , x3 , x4 ), the polynomial map F determined by this matrix is given
by:
x 7−→ (1 − x3 , −x2 − x3 − x4 − 1, −x1 + x2 x3 + x3 − 1, x3 + x4 )
The Jacobian of F at a = (−1, −1, 1, −1) is invertible, so F : M → N is a
bijection for some neighborhoods M and N of a and (0, 0, 0, 0), respectively, and
94
for all (m1 , m2 , m3 , m4 ) ∈ M , mi 6= 0. Let q = (q1 , q2 , q3 , q4 ) ∈ F4 ∩ N , so there
exists a solution b = (b1 , b2 , b3 , b4 ) ∈ M to the system F (b) = q. From the first and
fourth equation of the polynomial map, we have b3 , b4 ∈ F. This implies b2 ∈ F
from the second equation, and hence b1 ∈ F from the third equation. Therefore this
pattern is spectrally arbitrary over F by the surjective neighborhoods condition.
12. Consider the following pattern and realization.



 a
 ∗ ∗ 0 0 






 b
 ∗ ∗ ∗ 0 






 0
 0 ∗ ∗ ∗ 






0
0 0 ∗ 0

1 0 0 


c 1 0 


d e 1 


0 f 0
We set the polynomial map F determined by this realization equal to (q1 , q2 , q3 , q4 ) ∈
F and solve to get the following system of equations:
a + c + e + q1 = 0
ac + ae − b + ce − d − f − q2 = 0
(A.1)
−ace + ad + af + be + cf − q3 = 0
bf − acf − q4 = 0
Calculating a Groebner basis for the set of polynomials {a + c + e + q1 , ac + ae −
b + ce − d − f − q2 , −ace + ad + af + be + cf − q3 , bf − acf − q4 } with respect to
a, b, c, d, e, f and setting each element of the Groebner basis equal to zero, we get
95
the following system of equations.
a + c + e − q1 = 0
b + c2 + ce − cq1 + d + e2 − eq1 + f + q2 = 0
cd + 2de − dq1 + e3 − e2 q1 + 2ef + eq2 − f q1 + q3 = 0
(A.2)
2
2
df + e f − ef q1 + f + f q2 + q4 = 0
ce2 f − cef q1 + cf 2 + cf q2 + cq4 + e3 f − 2e2 f q1 + ef q12 + ef q2 +
2eq4 − f q1 q2 − f q3 − q1 q4 = 0
Solving system (2) (which is equivalent to solving system (1)) yields the following
solutions.
eq4 − e − q1 − q3
+ eq1 + q2 + q4 + 1
−e3 − 2e2 q1 − eq12 − eq2 − 2eq4 − q1 q2 + q3 − q1 q4
c=
e2 + eq1 + q2 + q4 + 1
a=
e2
d = −e2 − eq1 − q2 − q4 − 1
b=
(e2
1
(e4 + 3e3 q1 + e3 q3 + 3e2 q12 + e2 q2 +
+ eq1 + q2 + q4 + 1) 2
2e2 q1 q3 + e2 q2 q4 + 4e2 q4 + eq13 + eq1 q42 + 2eq1 q2 + eq12 q3 + eq2 q3
− eq3 + 5eq1 q4 + eq1 q2 q4 + 3eq3 q4 + q43 − q32 + 2q2 q42 + 2q42 + q12 q2
− q1 q3 + q1 q2 q3 + q12 q4 + q22 q4 + 2q2 q4 + q1 q3 q4 + q4 )
We can pick e ∈ F large enough such that a, b, c, d are all defined, nonzero, and lie
in F. Hence, this pattern is spectrally arbitrary over F.
96
13. Consider the following pattern

 0 ∗ ∗


 ∗ 0 ∗


 ∗ 0 ∗


∗ 0 0
and nilpotent realization.


0 
 0 1 2



 1
0 
 3 0 1



 2

 − 0 −1
∗ 
 3


∗
− 13 0 0

0 


0 


1 


1
The following realization yields a polynomial map F whose Jacobian is nonsingular
at a = 1, 13 , − 23 , − 13 .


 0 1 2 0 




 x2 0 1 0 




 x3 0 −1 1 




x4 0 0 x1
Thus, there exists neighborhoods M and N of a and (0, 0, 0, 0), respectively, such
that F : M → N is a bijection and for all (m1 , m2 , m3 , m4 ) ∈ M , mi 6= 0. Pick
(q1 , q2 , q3 , q4 ) ∈ F4 ∩ N and set F (x1 , x2 , x3 , x4 ) = (q1 , q2 , q3 , q4 ) to get the following
system of equations.
x1 = 1 − q 1
x4 =
−q13 + q2 q12 + 3q12 − 2q2 q1 − q3 q1 − 3q1 + q2 + q3 + q4 + 1
2q1 − 3
x3 = q12 − q2 q1 − q1 + q3 + 2x4
x2 = q1 − q2 − 2x3 − 1
We can pick each qi close enough to 0 to ensure that x1 , x2 , x3 , x4 are all nonzero.
Since (x1 , x2 , x3 , x4 ) ∈ F4 , this pattern is spectrally arbitrary over F by the surjective neighborhoods condition.
97
14. Consider the following pattern

 0 ∗ ∗


 ∗ ∗ 0


 ∗ 0 ∗


∗ 0 0
and nilpotent realization.


1 1
∗ 
 0



 81
0 
 − 20 −3 0



 16
 −
0 

 5 0 2


1
∗
0 0
4

1 


0 


0 


1
The following realization yields a polynomial map F whose Jacobian is nonsingular
at a = (1, 1, 1, 1).


x2 x3 x4 
 0


 81

 − 20 −3 0 0 


 16

 −

0
2
0
 5



1
0 0 x1
4
Thus, there exists neighborhoods M and N of a and (0, 0, 0, 0), respectively, such
that F : M → N is a bijection and for all (m1 , m2 , m3 , m4 ) ∈ M , mi 6= 0. Pick
(q1 , q2 , q3 , q4 ) ∈ F4 ∩ N and set F (x1 , x2 , x3 , x4 ) = (q1 , q2 , q3 , q4 ) to get the following
system of equations.
x1 = 1 − q 1
4 (q13 − q2 q12 − 3q12 + 2q2 q1 + q3 q1 + 3q1 − q2 − q3 − q4 − 1)
q12 − 3q1 − 4
1
x3 =
−q1 x4 + 4q12 − 4q2 q1 − 16q1 + 12q2 + 4q3 + 4x4 + 60
64
1
x2 =
(−20q1 + 20q2 − 64x3 + 5x4 + 140)
81
x4 =
We can pick each qi close enough to 0 to ensure that x1 , x2 , x3 , x4 are all nonzero.
Since (x1 , x2 , x3 , x4 ) ∈ F4 , this pattern is spectrally arbitrary over F by the surjective neighborhoods condition.
98
15. Consider the following pattern and nilpotent realization.



0
 0 1 0
 0 ∗ 0 0 






0
 −1 1 1
 ∗ ∗ ∗ 0 






 0 0 1
 0 0 ∗ ∗ 
1






0 1 −2 −2
0 ∗ ∗ ∗










We change four nonzero entries to indeterminants as follows.


 0 1 0 0 




 x1 1 1 0 




 0 0 1 1 




0 x2 x3 x4
The polynomial map F determined by this matrix has a nonsingular Jacobian at
(−1, 1, −2, −2), so there exists real neighborhoods M and N of (−1, 1, −2, −2)
and (0, 0, 0, 0), respectively, such that F : M → N is a bijection and for all
(m1 , m2 , m3 , m4 ) ∈ M , mi 6= 0. Pick (q1 , q2 , q3 , q4 ) ∈ Q4√+ ∩N and set F (x1 , x2 , x3 , x4 ) =
(q1 , q2 , q3 , q4 ) to get the following system of equations.
x1 =
p
1
−q1 − q2 ± (−q1 − q2 − 1) 2 − 4q4 − 1
2
x4 = −2 − q1
x3 = −2q1 − x1 − 3 − q2
x2 = q1 (−x1 ) − q1 − q2 − q3 − 2x1 − 1
We can pick each qi small enough so that (−q1 − q2 − 1) 2 −4q4 > 0, so that x1 ∈ R.
This implies x1 and consequently x2 , x3 , x4 will lie in Q√+ , so this pattern satisfies
the surjective neighborhoods condition over Q√+ .
99
16. Consider the following pattern

 ∗ ∗ 0


 0 ∗ ∗


 0 0 ∗


∗ ∗ 0
and nilpotent realization.


0 
 −2 1 0




1 1
∗ 
 0




 0
0 1
∗ 




2 −1 0
0

0 


3 


1 


0
The following matrix yields a nonsingular Jacobian polynomial map at a = (1, 1, 1, 3).


 −2 x1 0 0 




 0 x2 x3 x4 




 0

0
1
1




2 −1 0 0
The polynomial map F determined by this matrix has a nonsingular Jacobian at
a, so there exists neighborhoods real M and N of a and (0, 0, 0, 0), respectively,
such that F : M → N is a bijection and for all (m1 , m2 , m3 , m4 ) ∈ M , mi 6= 0.
Pick (q1 , q2 , q3 , q4 ) ∈ Q4√+ ∩ N and set F (x1 , x2 , x3 , x4 ) = (q1 , q2 , q3 , q4 ) to get the
following system of equations.
√
6q1 − 4q2 + q3 ± γ − 14
x1 =
4 (q1 − q2 − 3)
x2 = 1 − q 1
x4 = 3 − q 1 + q 2
x3 = −2q1 x1 + 2q2 x1 + 3q1 − q2 + q3 + 6x1 − 5
Where γ = 4q12 + 4q3 q1 + 4q4 q1 − 8q1 + q32 − 4q3 − 4q2 q4 − 12q4 + 4. We can pick each
qi small enough so that γ > 0, so that x1 ∈ R. This implies x1 and consequently
x2 , x3 , x4 will lie in Q√+ , so this pattern satisfies the surjective neighborhoods
condition over Q√+ .
100
17. Consider the following pattern

 ∗ ∗ 0


 0 ∗ ∗


 0 0 0


∗ ∗ 0
and nilpotent realization.


0 
 −2 1 0 0




∗ 
 0 1 1 − 32




 0 0 0 1
∗ 




∗
−4 2 0 1










The following realization yields a polynomial map F whose Jacobian at a =
(1, 1, 1, − 23 ) is nonsingular.


 −2 x1 0 0 




 0 x2 x3 x4 




 0 0 0 1 




−4 2 0 1
Thus, there exists real neighborhoods M and N of a and (0, 0, 0, 0), respectively,
such that F : M → N is a bijection and for all (m1 , m2 , m3 , m4 ) ∈ M , mi 6= 0.
Pick (q1 , q2 , q3 , q4 ) ∈ Q4√+ ∩ N and set F (x1 , x2 , x3 , x4 ) = (q1 , q2 , q3 , q4 ) to get the
following system of equations.
√
6q1 − 4q2 + q3 ± γ − 14
x1 =
4 (q1 − q2 − 3)
x2 = 1 − q 1
x4 =
x3 =
1
(q1 − q2 − 3)
2
1
(2q1 x1 − 2q2 x1 − 4q1 + 2q2 − q3 − 6x1 + 8)
2
Where γ = 4q12 + 4q3 q1 + 4q4 q1 − 8q1 + q32 − 4q3 − 4q2 q4 − 12q4 + 4. We can pick each
qi small enough so that γ > 0, so that x1 ∈ R. This implies x1 and consequently
101
x2 , x3 , x4 will lie in Q√+ , so this pattern satisfies the surjective neighborhoods
condition over Q√+ .
18. Consider the following pattern and realization.


 ∗ ∗ 0


 ∗ ∗ ∗


 0 0 ∗


∗ 0 0
0 


0 


∗ 


∗

 a b 0


 1 c 1


 0 0 d


e 0 0

0 


0 


1 


f
We set the polynomial map F (a, b, c, d, e, f ) determined by this realization equal
to (q1 , q2 , q3 , q4 ) ∈ Qβ to get the following system of equations.
a = −q1 − c − d − f
b = −q2 + ac + ad + af + cd + cf + df
3
d (−f ) − d2 f 2 − d2 f q1 − df 3 − df 2 q1 − df q2 + q4
e= 2
c + cd + cf + cq1 + d2 + df + dq1 + f 2 + f q1 + q2
!
r
√
q
3
1
2γ2
1
3
3
2
γ
+
4γ
+
γ
+
d = (−f − q1 ) − √
q
1
2
3
3
p
3
3 2
3
3
2
3
γ1 + 4γ2 + γ3
Where γ1 , γ2 , γ3 are given as follows.
γ1 = 20f 3 + 15q1 f 2 − 3q12 f + 18q2 f + 2q13 − 9q1 q2 − 27q3
γ2 = 2f 2 + q1 f − q12 + 3q2
γ3 = 20f 3 + 15q1 f 2 − 3q12 f + 18q2 f + 2q13 − 9q1 q2 − 27q3
We can pick c, f ∈ Q large enough so that a, b, d, e ∈ Qβ and are all nonzero.
Thus, this pattern is spectrally arbitrary over Qβ .
102
A.3
The Pattern B1
Consider the following pattern and nilpotent realization.

 ∗ ∗ 0


 ∗ 0 ∗
B1 = 

 0 0 ∗


∗ 0 0



 −1 − i


 −i



0


−1
0 


0 


∗ 


∗
1 0 0 


0 1 0 


0 i 1 


0 0 1
The following realization yields a polynomial map F whose Jacobian map at a =
(a1 , a2 , a3 , a4 ) = (−1 − i, −i, i, −1) is nonsingular.


 x1


 x2


 0


x4
1
0
0
1
0 x3
0
0
0 


0 


1 


1
Thus, there exists complex neighborhoods M and N of a and (0, 0, 0, 0), respectively,
such that F : M → N is a bijection and for all (m1 , m2 , m3 , m4 ) ∈ M , mi 6= 0. Pick
q = (q1 , q2 , q3 , q4 ) ∈ Qβ 4 ∩ N and set F (x1 , x2 , x3 , x4 ) = q to get the following system of
equations.
x1 = −x3 − q1 − 1
x2 = x3 (−q1 − 1) − x23 − q1 − q2 − 1
x4 = x23 (q1 + 1) + x3 (q1 + q2 + 1) + x33 − q4
0 = x23 (−q1 − 1) + x3 (−q1 − q2 − 1) − x33 − q1 − q2 − 1 − q4
Since the fourth equation is cubic and q1 , q2 , q3 , q4 ∈ Qβ , any solution x3 will also lie
103
in Qβ , and hence x1 , x2 , x3 ∈ Qβ . We can pick each qi small enough so that each xi 6= 0,
so this pattern satisfies the surjective neighborhoods condition over Qβ .
104
Appendix B
Sample Mathematica Code
The following code was used to determine that the 4×4 pattern B is spectrally arbitrary
over F3 .
T={};
Do[
AppendTo[T,
Expand[CharacteristicPolynomial[
{{a,1,0,0},{b,0,1,0},{c,0,0,1},{d,0,0,0}},x],Modulus -> 3]],
{a,0,2,1},{b,0,2,1},{c,0,2,1},{d,0,2,1}]
t={};
Do[
AppendTo[t,
Expand[CharacteristicPolynomial[
{{a,1,b,c},{d,e,1,f},{g,0,0,1},{h,i,0,0}},x],Modulus -> 3]],
{a,1,2,1},{b,1,2,1},{c,1,2,1},{d,1,2,1},{e,1,2,1},{f,1,2,1},
{g,1,2,1},{h,1,2,1},{i,1,2,1},{j,1,2,1},{k,1,2,1}]
t’=Union[t]; {Length}[t’]
T’=Union[T]; Complement[T’,t’]
105
This code creates a set T 0 of all possible characteristic polynomials of degree 4 over
F3 . The set t0 consists of all characteristic polynomials realized by the pattern B.
106
Appendix C
Sample Sage Code
The following code was used to determine that the 5 × 5 pattern C (see chapter 5.2.2)
is spectrally arbitrary over F3 .
K.<a>=GF(3)
S = set()
for a in K:
if a != 0:
for b in K:
if b != 0:
for c in K:
if c != 0:
for d in K:
if d != 0:
for e in K:
if e != 0:
for f in K:
if f != 0:
for g in K:
107
if g != 0:
for h in K:
if h != 0:
for i in K:
if i != 0:
for j in K:
if j != 0:
for k in K:
if k != 0:
for l in K:
if l != 0:
M = Matrix([[a,1,0,b,c],[0,d,1,e,0],[f,0,g,1,0],[0,0,0,h,1],[i,0,0,j,0]])
lens(S)
T = set()
for a in K:
for b in K:
for c in K:
for d in K:
N = Matrix([[a,1,0,0],[b,0,1,0],[c,0,0,1],[d,0,0,0]])
T.add(N.charpoly())
108
T-S
T is the set of all characteristic polynomials realized by C over F3 , and S is the set of
all possible 5 × 5 characteristic polynomials over F3 .
109
Bibliography
[1] T. Britz, J. J. McDonald, D. D. Olesky, and P. van den Driessche. Minimal spectrally
abritrary sign patterns. SIAM J. Matrix Anal. Appl., 26(2004), no. 1, 257–271.
[2] Rajesh Pereira. Nilpotent matrices and spectrally arbitrary sign patterns. Electronic Journal of Linear Algebra, Vol. 16 (2007), pp. 232–236.
[3] J. J. McDonald and A. A. Yielding. Complex spectrally arbitrary zero-nonzero
patterns, Linear Algebra and Multilinear Algebra, 3(60), 285–299.
[4] C. Garnett and B. L. Shader. The nilpotent-centralizer method for spectrally arbitrary patterns. Linear Algebra and its Applications, Volume 438, Issue 10, pp.
3836–3850, May 2013.
[5] C. Garnett and B. L. Shader. A proof of the Tn conjecture: Centralizers, Jacobians
and spectrally arbitrary sign patterns. Linear Algebra and its Applications, Volume
436, Issue 12, pp. 4451–4458, June 2012.
[6] L. Corpuz, J. J. McDonald. Spectrally Arbitrary Zero-Nonzero Patterns of Order
4, Linear Multilinear Algebra, 55(2007), no. 3, 249–273.
[7] J. H. Drew, C. R. Johnson, D. D. Olesky, and P. van den Driessche. Spectrally
arbitrary patterns. Linear Algebra Appl., 308(2000), 121–137.
110
[8] L. M. DeAlba, I. R. Hentzel, L. Hogben, J. J. McDonald, R. Mikkelson, O. Pryporova, B. Shader, and K. N. Vander Meulen, Spectrally arbitrary patterns: reducibility and the 2n conjecture for n = 5. Linear Algebra Appl., 423(2007), no.
2-3, 262-276.
[9] D. S. Dummit and R. M. Foote, Abstract Algebra, 3rd Ed., John Wiley and Sons,
Inc., 2004.
[10] M. S. Cavers and K. N. Vander Meulen. Spectrally and inertially arbitrary sign
patterns. Linear Algebra Appl., 394(2005), 53–72.
[11] K. Vander Meule and A. Van Tuyl. Zero-nonzero patterns for nilpotent matrices
over finite fields, Electronic J. Linear Algebra, Vol. 18 (2009), pp. 628-648.
[12] W. Rudin. Principles of Mathematical Analysis, 3rd Ed., McGraw-Hill Book Company, 1976.
[13] M. S. Cavers and S. M. Fallat. Allow problems concerning spectral properties of
patterns. Electronic Journal of Linear Algebra, Vol. 23 (2012), pp. 731–754.
[14] N. Campbell, K. N. Vander Meulen, and A. Van Tuyl. Structure of nilpotent matrices over fields. Electronic Journal of Linear Algebra, Vol. 22 (2011), pp. 931–958.
[15] E. J. Bodine and J. J. McDonald. Spectrally arbitrary zero-nonzero patterns of the
finite fields. Linear and Multilinear Algebra, 3(60), 285–299.
[16] R. A. Brualdi, L. Hogben, B. L. Shader, AIM Workshop: Spectra of families of matrices described by graphs, digraphs and sign patterns, Final Report: Mathematical
111
Results, March 30, 2007. Available at:
http://aimath.org/pastworkshops/matrixspectrumrep.pdf.
[17] H. Bergsma, K. N. VanderMeulen, and A. Van Tuyl. Potentially nilpotent patterns
and the nilpotent-Jacobian method. Linear Algebra and its Applications, Vol. 436
(2012), pp. 4433–4445
[18] E. J. Bodine. Spectrally arbitrary sign patterns over finite fields. Dissertation, Washington State University. 2010.
[19] W. Fulton. Algebraic Curves, 3rd Ed., William Fulton, 2008. Available at:
www.math.lsa.umich.edu/ wfulton/CurveBook.pdf
[20] M. Arav, F. Hall, Z. Li, K. Kaphle, and N. Manzagol. Spectrally arbitrary tree
sign patterns of order 4. Electronic Journal of Linear Algebra, Vol. 20 (2010), pp.
180–197
[21] R. A. Brualdi, H. J. Ryser. Combinatorial Matrix Theory. New York: Cambridge
University Press, Cambridge, 1991.
[22] S. Lang. Algebra. Addison-Wesley Publishing Company, Inc., Reading, Massachusetts, 1965.
[23] D. Cox, J. Little, and D. O’Shea. Ideals, Varieties, and Algorithms: An Introduction
to Computational Algebraic Geometry and Commutative Algebra. Springer-Verlag,
New York, 1992.
[24] M. Catral, D. D. Olesky, and P. van den Driessche. Allow problems concerning
112
spectral properties of sign pattern matrices: a survey, Linear Algebra and its Applications, 430 (2009), pp. 3080-3094.
[25] C. A. Eschenbach and Z. Li. Potentially nilpotent sign pattern matrices. Linear
Algebra and Its Applications, 299 (1999), pp. 81-99.
[26] B. L. Shader. Notes on the 2n-conjecture. Available at:
http://www.aimath.org/pastworkshops/matrixspectrum.html.
[27] T. Tao. Infinite fields, finite fields, and the Ax-Grothendieck theorem. Available at:
terrytao.wordpress.com/2009/03/07/infinite-fields-finite-fields-and-the-axgrothendieck-theorem/
[28] Weisstein, Eric W. “Constructible Number.” From MathWorld - A Wolfrom Web
Resource. Available at:
http://mathworld.wolfram.com/constructibleNumber.html