Topics in geometry, analysis
and inverse problems
Hans Rullgård
Department of Mathematics
Stockholm University
2003
Doctoral dissertation 2003
Department of Mathematics
Stockholm University
SE-106 91 Stockholm
Sweden
Typeset by LATEX
c 2003 by Hans Rullgård
ISBN 91-7265-738-3
Printed by Akademitryck AB, Edsbruk
Contents
Summary
I
7
Polynomial amoebas
11
1 Introduction
13
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.2 Outline of chapter . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2 The complex torus
17
3 Foundations
22
3.1 Some classical results . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.2 A class of almost hypergeometric functions . . . . . . . . . . . . 26
3.3 Functorial properties . . . . . . . . . . . . . . . . . . . . . . . . . 30
4 Patchworking amoebas
31
4.1 Dual polyhedral subdivisions . . . . . . . . . . . . . . . . . . . . 32
4.2 The spine of an amoeba . . . . . . . . . . . . . . . . . . . . . . . 33
5 The complement components of an amoeba
35
6 Convexity of the Ronkin function
40
6.1 The Monge-Ampère operator . . . . . . . . . . . . . . . . . . . . 40
6.2 Monge-Ampère measures on amoebas . . . . . . . . . . . . . . . 44
7 Amoebas and real algebraic geometry
8 Amoebas of varieties of codimension greater
8.1 The amoeba complement . . . . . . . . . . .
8.2 The Ronkin function . . . . . . . . . . . . . .
8.3 The spine . . . . . . . . . . . . . . . . . . . .
8.4 The Newton polytope . . . . . . . . . . . . .
50
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51
51
52
53
53
9 Examples of amoebas
55
10 Some open problems
60
11 List of notations
62
II
67
Differential equations in the complex plane
3
1 Some general remarks
69
2 On polynomial eigenfunctions for a class of differential operators
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Calculation of the matrix . . . . . . . . . . . . . . . . . . . . . .
2.3 Probability measures whose Cauchy transform satisfies an algebraic equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 Root measures and the Cauchy transform . . . . . . . . . . . . .
2.5 Root measures of eigenpolynomials . . . . . . . . . . . . . . . . .
III
Radon transforms and tomography
70
70
73
77
83
85
91
1 Introduction
93
2 Background
93
2.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
2.2 Important problems . . . . . . . . . . . . . . . . . . . . . . . . . 94
2.3 Known results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
3 Preliminaries
96
3.1 Function spaces and norms . . . . . . . . . . . . . . . . . . . . . 96
3.2 Estimates of operator norms . . . . . . . . . . . . . . . . . . . . . 98
3.3 Radon transforms and Fourier transforms . . . . . . . . . . . . . 100
4 Forward estimates
102
5 Stability of the inverse problem
5.1 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Some sufficient conditions for a weight function to be in W k . . .
5.3 Proof of Theorem 3 . . . . . . . . . . . . . . . . . . . . . . . . . .
107
107
109
110
6 Inversion formula for
6.1 Main results . . . .
6.2 Proofs. . . . . . . .
6.3 Numerical test. . .
the exponential Radon
. . . . . . . . . . . . . . .
. . . . . . . . . . . . . . .
. . . . . . . . . . . . . . .
4
transform
115
. . . . . . . . . . . 115
. . . . . . . . . . . 117
. . . . . . . . . . . 120
Acknowledgements
During my work on this thesis, I have benefited from the help of many people.
I would, in particular, like to thank my thesis advisor Mikael Passare, Grigory
Mikhalkin and Tanja Bergkvist, who I have had the pleasure to work together
with on different projects. I also want to thank everybody at the Department
of Mathematics at Stockholm University. It has been a great time here.
5
6
Summary
This thesis consists of three independent parts, dealing with different aspects of
real and complex analysis and geometry.
Part I deals with amoebas of polynomials in several complex variables, a
concept introduced by Israel Gelfand, Mikhail Kapranov and Andrei Zelevinsky.
Let f be a Laurent polynomial in n complex variables, and let C∗ = C r {0}.
Define the mapping Log : Cn∗ → Rn by Log(z1 , . . . , zn ) = (log |z1 |, . . . , log |zn |).
Then the amoeba of f is defined to be the set Af = Log(f −1 (0)). One of
the motivations for considering this concept was certain relations between the
amoeba and the Newton polytope of the polynomial. These relations have
been further investigated by several researchers, and it has been found, for
example, that the connected components of the amoeba complement correspond
to different integer points in the Newton polytope of the polynomial. One
consequence of this correspondence is that the number of connected components
can be bounded from below and from above by the number of vertices of the
Newton polytope and the number of integer points in the Newton polytope
respectively.
The first part of the thesis comprises work on amoebas carried out by the
author in collaboration with Mikael Passare and Grigory Mikhalkin. We also
give a summary of some results by other researchers. Our main results are:
• For any polynomial f we construct a polyhedral complex approximating
the amoeba of f , which we call the spine of the amoeba. The spine may be
computed by means of certain hypergeometric functions in the coefficients
of f . (See sections 3.2 and 4.2.)
• We show that for polynomials with a prescribed Newton polytope, the
number of connected components in the amoeba complement can take
any integer value between the upper and lower bounds mentioned above.
(See section 5.)
• For polynomials of two variables, we show that the area of the amoeba
is not greater than π 2 times the area of the Newton polytope, and we
characterize the polynomials for which the maximal area is obtained. (See
sections 6.2 and 7.)
We also comment on some possibilities of extending the theory to subvarieties
of Cn∗ of codimension greater than 1.
Part II is concerned with the asymptotic zero distribution of certain sequences
of polynomials in one complex variable, whose degrees tend to infinity. Specifically, let k be an integer and let Q0 , . . . , Qk be polynomials in one complex
7
variable, with deg Qj ≤ j and with Qk monic of degree k. Then the differential
operator
k
X
TQ f =
Qj f (j)
j=0
has a unique monic eigenpolynomial pn of degree n for all sufficiently large
integers n. We obtain the following results on the asymptotic zero distribution
of the polynomials pn .
• In the limit, when n → ∞, the zeros of pn are distributed according to a
certain compactly supported probability measure µ in the complex plane.
• The probability measure µ is the uniqueZ compactly supported positive
dµ(ζ)
measure whose Cauchy transform C(z) =
satisfies Qk (z)C(z)k =
z−ζ
1 for almost all z ∈ C.
• The support of µ is the union of finitely many curve segments, and both
the support and its complement are connected.
These results answer certain questions posed by Gisli Masson and Boris
Shapiro. The work was carried out by the author in collaboration with Tanja
Bergkvist.
Part III addresses some problems related to inversion of generalized Radon
transforms in two dimensions. Let f (x) be a function defined in R2 and let
ρ(θ, x) be a function on S 1 × R2 . Then
Z
Rρ f (θ, s) =
ρ(θ, x)f (x) dl
x·θ=s
is called the weighted Radon transform of f with weight ρ. Here (θ, s) ∈ S 1 × R
and dl denotes linear Lebesgue measure on the line {x ∈ R2 ; x · θ = s}.
Two special kinds of weight functions ρ are of particular interest in applications. If µ is a real number, and
ρ(θ, x) = exp(µx · θ ⊥ )
where θ⊥ denotes the unit vector in S 1 obtained by rotating θ counter-clockwise
through a right angle, then Rρ is called an exponential Radon transform. We
denote the exponential Radon transform also by Rµ . The other special case is
when µ(x) is a compactly supported function defined in R2 and
Z ∞
ρ(θ, x) = exp −
µ(x + tθ⊥ ) dt .
0
8
In this case Rρ is called an attenuated Radon transform, and we also use the
notation Rµ . The function µ is called an attenuation function.
The problem we are interested in is how f can be computed when ρ and Rρ f
are known. In particular, we want to know if this problem can be solved when
1
Rρ f is known only on a subset of S 1 × R, for example the subset S+
× R, where
1
S+
denotes one half of the unit circle. This is of practical interest, for example
in applications to SPECT (Single Photon Emission Computed Tomography).
We obtain, among other things, the following results.
• We prove the following stability result for attenuated Radon transforms.
Let µ be a Hölder continuous attenuation function, and let K ⊂ R2 be
a compact set. Let kf kk denote the Sobolev norm with exponent k (see
Section 3.1) and let kgk denote the L2 norm of a function on S 1 × R.
Then there exists a constant C, depending only on µ and K, such that
kf k−1/2 ≤ CkRµ f |S+1 ×R k
for all functions f with support in K (see Theorem 5). This implies, in
particular, that f can be determined with arbitrary precision if sufficiently
good approximations of µ and Rµ f are known (see Corollary 2).
• We derive an essentially explicit inversion formula for the exponential
1
Radon transform, which uses only the restriction of Rµ f to S+
× R (see
Corollary 6).
Publications
[1] H. Rullgård: Stratification des espaces de polynômes de Laurent et structure
de leurs amibes, C. R. Acad. Sci. 2000, 355–358.
[2] M. Passare,
H. Rullgård:
Amoebas,
Monge-Ampère measures,
and triangulations of the Newton polytope. Preprint:
http://www.math.su.se/reports/2000/10. To appear in Duke Math.
J.
[3] G. Mikhalkin, H. Rullgård: Amoebas of maximal area, International Mathematics Research Notices, 9 (2001), 441–451.
[4] T. Bergkvist, H. Rullgård: On polynomial eigenfunctions for a class of differential operators, Math. Research Letters 9 (2002), 153–171.
[5] H. Rullgård:
An explicit inversion formula for the exponential Radon transform using data from 180◦ . Preprint:
http://www.math.su.se/reports/2002/9. To appear in Ark. Mat.
9
I
Polynomial amoebas
I
Polynomial amoebas
1
1.1
Introduction
Background
Consider the polynomial f (z) = z 2 + z − 2 = (z − 1)(z + 2). Suppose we are
interested in expanding the rational function 1/f (z) in a Laurent series, that is,
an infinite linear combination of Laurent monomials z k where k is an integer. It
is not difficult to find such a Laurent series expansion: By the geometric series
(1 − ζ)−1 = 1 + ζ + ζ 2 + . . . we have
+∞ 2
1X
1
1
1
=
−
=
·
2
f (z)
−2 1 − z 2+z
2 j=0
z +z
2
j
.
When the terms in the sum on the right are expanded, we see that only terms
where k/2 ≤ j ≤ k contain the monomial z k . So by expanding all terms and
collecting
finitely many monomials with the same exponent k, we obtain
Pthe
+∞
a series k=0 ak z k . It is not difficult to show that this series converges when
|z| < 1, and is then equal to 1/f (z).
Another expansion is obtained from the computation
j
+∞ 1
1
1
1 X 2
1
.
= 2·
−
=
f (z)
z 1 − 2−z
z 2 j=0 z 2
z
z2
This time we obtain, after expanding the terms in the sum, a series containing
only negative powers of z. Some computations show that this series converges
when |z| > 2 and is then equal to 1/f (z).
Are there any other Laurent series expansions of 1/f (z)? Observing that the
two expansions we have found so far were obtained by dividing out the constant
term and the z 2 -term respectively, and then using a geometric series, we try the
same trick with the z-term:
j
+∞ 1
1X 2
1
1
=
= ·
−
z
.
2
f (z)
z 1 − 2−z
z j=0 z
z
This time we run into a difficulty which was not encountered in the previous
two computations. When the terms in the sum are expanded, we find that,
for example, a nonzero constant term is present whenever j is even. In order
to know whether the sum represents a Laurent series (let alone a convergent
Laurent series) we would have to check whether the sum of all these constant
terms converges, and similarly for all other powers of z.
A better understanding can be gained by regarding the problem from a
geometric point of view. The function 1/f (z) has two poles at z = 1 and z = −2.
Hence it is holomorphic in the disc |z| < 1 and in the unbounded annulus |z| > 2,
and this is precisely where the first two series we computed converge. It is also
13
I
Polynomial amoebas
holomorphic in the annulus 1 < |z| < 2, and by a familiar theorem in the
theory of holomorphic functions,
it follows that 1/f (z) is represented there by
P+∞
a Laurent series f (z) = k=−∞ ak z k where
1
ak =
2πi
Z
|z|=r
z −k−1 dz
f (z)
and 1 < r < 2.
From the geometric picture we thus see immediately that there are precisely
three convergent Laurent series expansions of 1/f (z). Moreover, we have seen
how two of these can be computed explicitly, to whatever degree we like, by a
kind of geometric series trick. More generally, if f (z) is any polynomial in one
variable, then the number of convergent Laurent series expansions of 1/f (z) is
equal to one plus the maximal number of nonzero roots of f having mutually
distinct absolute values.
Let us now consider
P polynomials in several variables. Such a polynomial can
be written f (z) = α∈A cα z α . Here α = (α1 , . . . , αn ) ∈ Zn are multiorders,
z α = z1α1 . . . znαn and cα are arbitrary complex numbers. The summation takes
place over a finite subset A of the lattice Zn . (Actually, if f is a polynomial, all
the components αj must be positive, but it is just as natural here to allow f to
be a Laurent polynomial where the variables may be raised to negative powers.)
Singling out one of the points in A, say β, we try to write the series
!j
+∞
X cα
1
1
1
1 X
α−β
P
(1)
=
·
z
−
=
f (z)
cβ z β 1 + α∈A0 ccαβ z α−β
cβ z β j=0
cβ
0
α∈A
where A0 = A r {β}. How can we know if the sum on the right, when each term
is expanded, can be rearranged as a Laurent series? Let us say that the sum (1)
is well behaved if every monomial z ν occurs only in the expansions of a finite
number of terms
!j
X cα
α−β
−
.
z
cβ
0
α∈A
The problem of deciding whether the sum (1) is well behaved can easily be
understood from a geometric consideration. Imagine that we plot the points
in A in Euclidean space. The convex hull of these points is called the Newton
polytope of f . For example, if f (z1 , z2 ) = 1 + z15 + 80z12z2 + z13 z2 + 40z13 z22 + z13 z24 ,
then A = {(0, 0), (5, 0), (2, 1), (3, 1), (3, 2), (3, 4)} and the Newton polytope of f
is the triangle shown on the left in Figure 1. The significance of the Newton
polytope here is that the sum (1) is well behaved if and only if β is a vertex of
the Newton polytope. In our example, the sum is well behaved if β is one of
the points (0, 0), (5, 0) or (3, 4), which are corners of the triangle, but not if β is
one of the other three points. If f is a polynomial in one variable, the Newton
polytope is just a segment, and the vertices are its endpoints. These correspond
14
I
Polynomial amoebas
to the monomials of lowest and highest degree in f respectively. Our first two
computations succeeded, because there our β was one of these endpoints, but
the third computation ran into trouble (and would in fact have failed if we had
carried on) since β was a point inside the segment.
Let us now carry over the other way of understanding Laurent series expansions to polynomials of several variables. The multidimensional analogue
of an annulus is a circular domain {z ∈ Cn ; (|z1 |, . . . , |zn |) ∈ E} where E is
an open connected subset of Rn>0 . If a function is holomorphic in such a circular
then it can be represented there by a convergent Laurent series
P domain,
α
a
z
.
In
the special case of power series with only positive exponents,
α∈Zn α
the domain of convergence is a Reinhardt domain. These topics were studied
nearly a century ago by Georg Faber [7] and Fritz Hartogs [14]. For various
reasons, it is more convenient to describe a circular domain in the following
way. Let Log(z1 , . . . , zn ) = (log |z1 |, . . . , log |zn |), so that Log is a mapping
from (C r {0})n to Rn . Then a circular domain can be written in the form
Log−1 (E) = {z; Log(z) ∈ E} where E is an open connected set in Rn . Notice
that if one of the coordinates, say z1 , is replaced by z1−1 , which is quite natural
when we are dealing with Laurent series, the set E will simply be reflected in
the plane x1 = 0. Another nice property of this convention is that we need
only consider sets E which are convex. In fact, it can be shown that if E is
connected, then every function which is holomorphic in Log−1 (E) can automatically be extended to Log−1 (conv E) where conv E denotes the convex hull of
E.
When f is a polynomial in more than one variable, the singularities of 1/f ,
which must be avoided by Log−1 (E), are not discrete but spread out along the
surface of complex dimension n − 1 where f (z) = 0. The image of this surface
under the mapping Log is called the amoeba of f . Each connected component
E of the complement of the amoeba corresponds to a circular domain Log−1 (E)
where f (z) is never zero and hence 1/f (z) is holomorphic. We conclude that
there is a one-to-one correspondence between the connected components of the
complement of the amoeba and the convergent Laurent series expansions of
1/f (z).
Figure 1 shows the amoeba of the polynomial f (z1 , z2 ) = 1 + z15 + 80z12 z2 +
3
z1 z2 + 40z13z22 + z13 z24 . The three “tentacles” of the amoeba extend outside the
picture to infinity. There are five clearly visible complement components of the
amoeba in the picture. Two of these are bounded, while the remaining three
are infinitely large. These three unbounded regions are where the three well
behaved Laurent series which we found by the geometric series computation
converge.
Notice that there seems to be a kind of duality between the amoeba and the
Newton polytope. This feeling becomes even stronger if the Newton polytope is
triangulated by drawing lines as in the second picture. A more precise statement
is that the amoeba looks like a thickened graph, whose edges are perpendicular
to certain edges in the triangulation of the Newton polytope.
15
I
Polynomial amoebas
6
6
5
5
4
4
3
3
2
2
1
1
-1
1
2
3
4
5
6
-1
-1
1
2
3
4
5
6
-1
Figure 1: Newton polytope, triangulated Newton polytope and amoeba of the
polynomial f (z) = 1 + z15 + 80z12 z2 + z13 z2 + 40z13z22 + z13 z24
It is almost obvious from looking at these pictures that each unbounded
component of the amoeba complement is associated to a vertex of the Newton
polytope. One might also surmise, that the two bounded components belong to
the points (2, 1) and (3, 2) in the Newton polytope, corresponding to the terms
80z12 z2 and 40z13 z22 in f (z). This is actually the case, in a sense made precise by
the concept of the order of a complement component introduced by Forsberg,
Passare and Tsikh. Notice that there seems to be no complement component
corresponding to the term z13 z2 . It is tempting to attribute this to the fact that
its coefficient is so much smaller than the coefficients of z12 z2 and z13 z22 . There is
some truth to this statement; a connection between the size of a coefficient and
the existence and size of a corresponding complement component does exist.
However, the connection is weaker than one might initially be lead to hope (see
Example 7 in section 9).
1.2
Outline of chapter
There are three main problems with which this chapter is concerned. These are
treated in sections 4, 5 and 6. The treatment is based on the papers [24], [25]
and [30], where many of the results of the thesis can also be found.
In section 2 we give the definitions of the objects we will be dealing with,
together with a discussion of the complex torus which is the space in which these
objects live. Section 3 outlines some results, to the most part taken from [9],
[12] and [29], which are of fundamental importance in the following treatment.
We also introduce certain functions, which are very nearly hypergeometric in
the GKZ sense, which turn out to have several interesting connections with
amoebas.
After these foundations have been established, we explore in section 4 the
duality between the amoeba and the Newton polytope. Certain aspects of this
duality were found in [8], [9] and [12]. We obtain, in section 4, another manifestation of this duality by exploiting a special convex function Nf associated to
the polynomial f . The idea of using this function in the study of amoebas comes
16
I
Polynomial amoebas
from the paper [29] It is defined by letting Nf (x) be the average of log |f (z)|
as z runs through Log−1 (x). This function encodes information both about the
amoeba (Theorem 1) and the Newton polytope (Theorem 2).
In section 5 we consider the problem of finding the number of complement
components of the amoeba of a given polynomial. Lower and upper bounds on
this number, in terms of the Newton polytope, were found in [12] and [9]. We
show that these bounds are sharp and also give some partial answers to certain
related problems.
Finally, in section 6 we study the convexity of the function Nf by means of
the Monge-Ampère operator, and find several relations between this function
and the hypersurface f −1 (0). One rather remarkable consequence is that for
polynomials of two variables, the area of the amoeba is no greater than π 2
times the area of the Newton polytope. Moreover, as was discovered jointly
with Mikhalkin, the polynomials for which the amoeba has maximal area are
those defining so-called Harnack curves which arise in real algebraic geometry.
The connection with real algebraic curves is outlined in section 7.
A possible generalization of the concepts treated in the thesis is outlined
in section 8. Section 9 contains some explicit examples, and in section 10 we
present a few open problems.
2
The complex torus
In this section we will define several objects related to a Laurent polynomial
which will be our main interest in later sections. First we shall discuss the
space where all these objects live, the complex torus.
The complex torus could simply be defined as the product space Cn∗ , where
C∗ = C r {0} is the multiplicative group of the complex field and n is a positive
integer. However, it is more natural to state the definitions in coordinate free
manner. Let therefore L be an n-dimensional lattice and L∗ = HomZ (L, Z) its
dual. The complex torus associated to L is the abelian group LC∗ = L ⊗ C∗
(where the tensor product is taken over Z). By choosing a basis for L, we obtain
an isomorphism between LC∗ and Cn∗ . In particular, we see that LC∗ has the
structure of a complex manifold as well as an abelian group. Wherever it is
convenient, we shall assume that such a basis has been chosen so that expressions
may be written in terms of the coordinate functions on Cn∗ . However, we shall
here establish notations which make it possible to avoid a particular choice of
basis most of the time.
The homomorphism ζ 7→ log |ζ| from C∗ to R determines a mapping Log =
id ⊗ log | · | : LC∗ → LR = L ⊗ R. If a basis is chosen for L, this mapping can
be written explicitly Log(z1 , . . . , zn ) = (log |z1 |, . . . , log |zn |).
Since Log−1 (0) is a compact subgroup of LC∗ , it has a unique Haar measure
η0 which is translation invariant and has total mass 1. If x ∈ LR and z ∈
17
I
Polynomial amoebas
Log−1 (x), then multiplication by z maps Log−1 (0) to Log−1 (x). Any two such
mappings differ only by a translation in Log−1 (0) and hence the direct images
of η0 under all such mappings coincide and therefore define a measure ηx on
Log−1 (x). We shall drop the subscript and denote by η the probability measure
ηx on any fiber Log−1 (x). The measure η can be computed by integrating the
differential form
1
dz1 ∧ . . . ∧ dzn
·
.
n
(2πi)
z1 . . . zn
Every α ∈ L∗ determines a function α ⊗ id : LC∗ → Z ⊗ C∗ = C∗ . This
function is usually denoted z 7→ z α and called a Laurent monomial. Similarly,
every a ∈ L gives rise to a function ζ 7→ ζ a := a⊗ζ from C∗ to LC∗ . If e1 , . . . , en
is a basis for L, then an isomorphism between LC∗ and Cn∗ is given explicitly by
z 7→ (z e1 , . . . , z en ). A finite linear combination of Laurent monomials is called
a Laurent polynomial. If f is a Laurent P
polynomial, we will let fα denote the
α
coefficient for z α in f . Hence, f (z) =
α∈L∗ fα z . The set of all Laurent
polynomials clearly is a C-algebra, isomorphic to the group algebra C[L∗ ].
P
A Laurent series is a (formal) linear combination f (z) = α∈L∗ fα z α where
there may be infinitely many nonzero terms in the sum. The Laurent series is
said to converge at z if the sum is absolutely convergent. Its domain of convergence is the largest open set in which it converges at every point, and f is
called convergent if its domain of convergence is nonempty. If f is a convergent
Laurent series, it is well known that its domain of convergence is of the form
Log−1 (Ω) where Ω ⊂ LR is a convex open set. A convergent Laurent series defines a holomorphic function on its domain of convergence. Conversely, if f (z)
is a holomorphic function defined in Log−1 (Ω) where Ω is open and connected,
then f (z) is represented by a Laurent series whose domain of convergence contains Log−1 (Ω). The coefficients of the Laurent series are given by
Z
fα =
z −α f (z) dη(z)
Log−1 (x)
for any x ∈ Ω. If f and g are convergent Laurent series, their sum f + g need
not be convergent and their product need not even be defined. If, however, their
domains of convergence have nonempty intersection, then f + g and f · g are
both holomorphic functions in the intersection of the domains of convergence of
f and g, and hence define convergent Laurent series.
Definition 1 (Gelfand, Kapranov, Zelevinsky [12]). If f is a Laurent
polynomial or a convergent Laurent series, then the amoeba of f , denoted A f ,
is the set Log(f −1 (0)).
If the domain of convergence of f is Log−1 (Ω), the amoeba of f should be
considered as a subset of Ω. We will write Acf = Ω r Af for the complement of
the amoeba in Ω.
18
I
Polynomial amoebas
Definition 2. If f is a Laurent series, Pf will denote the convex hull in L∗R of
the set {α ∈ L∗ ; fα 6= 0}. If f is a Laurent polynomial, Pf is called the Newton
polytope of f .
Definition 3 (Ronkin [29]). If f is a Laurent polynomial (or a Laurent series
converging in Log−1 (Ω)), the function Nf is defined in LR (or in Ω) by
Z
log |f (z)| dη(z).
(2)
Nf (x) =
Log−1 (x)
We will call Nf the Ronkin function of f .
A linear mapping T : L → M between two lattices, not necessarily of the
same dimension, induces in an obvious way mappings T ∗ : C[M ∗ ] → C[L∗ ],
TR : LR → MR and TC∗ : LC∗ → MC∗ . It is easily verified that
T ∗ f (z) = f (TC∗ z)
(3)
TR (Log z) = Log(TC∗ z)
(4)
for any z ∈ LC∗ and f ∈ C[M ∗ ]. If moreover T ∗ : M ∗ → L∗ is injective, then
both TR and TC∗ are surjective. In this case,
T C∗ η L = η M
(5)
where ηL and ηM denote the Haar measures on the fibers of Log : LC∗ → LR
and Log : MC∗ → MR respectively.
Let a ∈ L. The mapping ζ 7→ ζ a from C∗ to LC∗ induces a homomorphism
of homology groups H1 (C∗ , Z) → H1 (LC∗ , Z). The generator of H1 (C∗ , Z)
represented by the unit circle with its usual orientation, is mapped by this
homomorphism to an element in H1 (LC∗ , Z) which we will denote ρ1 (a). It is
not difficult to see that ρ1 : L → H1 (LC∗ , Z) is a homomorphism. Similarly,
any α ∈ L∗ defines an element ρ1 (α) in the cohomology group H 1 (LC∗ , Z)
via the mapping z 7→ z α . In de Rham cohomology, ρ1 (α) is represented by
the differential form (2πi)−1 z −α dz α = (2πi)−1 d log z α . For every pair a ∈
L, α ∈ L∗ , it is easy to see that hρ1 (α), ρ1 (a)i = hα, ai. Since H1 (LC∗ , Z)
and H 1 (LC∗ , Z) are free abelian groups of rank n, it follows that ρ1 : L →
H1 (LC∗ , Z) and ρ1 : L∗ → H 1 (LC∗ , Z) are in fact isomorphisms.
For any k = 1, . . . , n there is a Künneth map
H1 (LC∗ , Z)⊗k → Hk ((LC∗ )k , Z).
Together with the homomorphism induced in homology by the multiplication
map (LC∗ )k → LC∗ this defines a homomorphism H1 (LC∗ , Z)⊗k → Hk (LC∗ , Z)
19
I
Polynomial amoebas
which is easily seen V
to be alternating. Hence, composition with ρ1 defines a
k
homomorphism ρk :
L → HkL
(LC∗ , Z). Similarly, the multiplication in the
cohomology ring H ∗ (LC∗ , Z) =
H k (LC∗ , Z) together with ρ1 determines a
Vk ∗
k
k
homomorphism ρ :
L → H (LC∗ , Z). In de Rham cohomology, ρk (α1 ∧
. . . ∧ αk ) is represented by the differential form (2πi)−k z −α1 −...−αk dz α1 ∧ . . . ∧
dz αk . The mappings ρk and ρk are all isomorphisms, and satisfy hρk (α), ρk (a)i =
Vk
Vk ∗
hα, ai for all a ∈
L, α ∈
L .
If E is a contractible subset of LR , then there are canonical isomorphisms
Hk (LC∗ , Z) ∼
= Hk (Log−1 (E), Z) and H k (LC∗ , Z) ∼
= H k (Log−1 (E), Z), and we
may therefore identify these homology and cohomology groups for any E.
Let f be a Laurent polynomial or a convergent Laurent series and let E be a
connected component of Acf . We shall see that any such component is convex,
so in particular, it is contractible. Then f defines a mapping from Log−1 (E) to
C∗ , and hence a homomorphism f ∗ : H 1 (C∗ , Z) → H 1 (Log−1 (E), Z). Let ω be
the generator of H 1 (C∗ , Z) represented by the differential form (2πiζ)−1 dζ.
Definition 4 (Forsberg, Passare, Tsikh [9]). The vector α = ρ1
called the order of the complement component E.
−1
(f ∗ ω) is
An alternative definition is that the order is the unique α ∈ L∗ such that
the mappings z 7→ f (z) and z 7→ z α from Log−1 (E) to C∗ are homotopic. The
following relations are useful for computations with orders. If α is the order of
the complement component E and a ∈ L, then
Z
1
hα, ai = hf ∗ ω, ρ1 (a)i =
d log f (ζ a z).
2πi |ζ|=1
where z is any point in Log−1 (E). In particular the coordinates of α with
respect to some basis of L∗ are given by
Z
1
αj =
d log f (z1 , . . . , ζzj , . . . , zn ), z ∈ Log−1 (E)
2πi |ζ|=1
Z
(6)
zj ∂f /∂zj
dη(z), x ∈ E.
=
f (z)
Log−1 (x)
∗
A
If A is a subset of L
P , we willαlet C denote the set of all Laurent polynomials
of the form f (z) = α∈A fα z . Usually A will be a finite set. Thus CA is a
complex vector space whose points are Laurent polynomials. There is a natural
choice of coordinates on this space, namely the coefficients fα . The space CA
can be treated just as any other complex manifold. For example, we will consider
holomorphic functions defined on CA . The fact that f is used both to denote a
point in CA and a function on LC∗ is not likely to cause much confusion.
20
I
Polynomial amoebas
This is a natural place to establish some terminology and notations concerning
toric varieties which will be needed in sections 7 and 8. The following paragraphs
are not needed for the main part of the thesis. For an extensive treatment of
toric varieties, we refer to [6] and [4].
Let σ be a cone in L∗R generated by finitely many vectors in L∗ . Then
C[σ ∩ L∗ ] is the subalgebra of C[L∗ ] generated by all monomials z α with α ∈
σ ∩ L∗ . The affine toric variety associated to σ is defined to be the maximal
spectrum of C[σ ∩ L∗ ], and will be denoted Xσ . This variety can be given a
concrete realization as follows. Take a set of generators α1 , . . . , αk for σ and
consider the mapping φ : LC∗ → Ck defined by φ(z) = (z α1 , . . . , z αk ). Then
Xσ is isomorphic to the closure of φ(LC∗ ) in Ck (with the metric or Zariski
topology, whichever one prefers). If σ ⊂ τ are cones, then there is a natural
injective mapping Xτ → Xσ .
A general toric variety is constructed by gluing together affine toric varieties.
To keep track of the operations it is useful to introduce the notion of a fan.
A fan Σ in a real vector space V is a finite collection of polyhedral cones
σ ⊂ V , such that (i) if σ, τ ∈ Σ, then σ ∩ τ ∈ Σ and (ii) if σ ∈ Σ and τ is
any cone contained in σ, then τ ∈ Σ precisely if τ is a face of σ. A fan is
said to be complete if the union of all its cones is the entire vector space in
which it lives. If C is a cone in V , then the dual of C is defined to be the cone
C ∨ = {ξ ∈ V ∗ ; hξ, xi ≤ 0, ∀x ∈ C}. If C is generated by finitely many vectors
from a lattice in V , then its dual is generated by finitely many vectors of the
dual lattice
Suppose now that Σ is a fan in LR whose cones are generated by finitely
many lattice vectors. For every σ ∈ Σ, consider the affine toric variety X σ∨ .
From the disjoint union of all these varieties we form a new variety by identifying
Xτ ∨ with its image in Xσ∨ whenever τ ⊂ σ are in Σ. The variety so obtained
is called a toric variety and will be denoted XΣ . The affine toric variety arising
from the zero cone σ = {0} deserves special attention. The dual of σ is the
whole space L∗R and its associated affine toric variety Xσ∨ is isomorphic to the
complex torus LC∗ . This space is contained as an open dense subset in XΣ .
The toric variety XΣ is compact if and only if Σ is complete.
Now let P be a polytope in L∗R with vertices in L∗Q . If F is a face of P , then
the normal cone to P at F is defined to be the cone nc(F, P ) = {x; hξ − η, xi ≤
0, ∀ξ ∈ P, η ∈ F }. The cone dual to nc(F, P ) is cone(F, P ) = {t(ξ − η); t ≥
0, ξ ∈ P, η ∈ F }, which can be thought of as the cone of all vectors pointing
from F into P . The collection of all normal cones, as F runs over all faces of
P , is a complete fan Σ, and hence defines a compact toric variety XP = XΣ . If
F is any face of P then we write X(F ) = Xcone(F,P ) and let V (F ) denote the
closure of X(F ) r ∪G⊃F X(G) where the union is taken over all faces G properly
containing F . This system of subvarieties of XP looks like the polytope P , in
the sense that dim V (F ) = dim F and V (F ) ⊂ V (G) precisely if F ⊂ G. We
remark that there is a mapping, known as the moment map, from XP to P ,
taking V (F ) onto F for every face F of P . If f is a Laurent polynomial whose
21
I
Polynomial amoebas
Newton polytope is P , then f defines a closed hypersurface V ⊂ XP . The image
of V under the moment map was called the compactified amoeba in [12].
Two familiar examples of toric varieties are obtained by taking the polytope
P to be the standard simplex conv(0, e1 , . . . , en ) or the unit cube [0, 1]n in Rn .
In the first case, XP is the projective space Pn . For example, if n = 2, then P
is a triangle and XP is the projective plane. When F is a side of the triangle
P , V (F ) is one of the coordinate axes or the line at infinity. In the second case,
XP is the product P1 × . . . × P1 of n copies of the Riemann sphere.
3
Foundations
This section contains a collection of results which are fundamental for the study
of polynomials from the amoeba point of view. A major part of the material
has appeared earlier in [8], [9], [12] and [29]. However, it seems that much is
gained by studying the interplay between different objects, namely the Newton
polytope, the amoeba and the convex function Nf , which have not been considered all at once in these earlier works. For this reason, a selection of previously
known results are presented here in an attempt to carry out a unified approach
to the subject. One new aspect in this presentation is that certain results are
generalized from Laurent polynomials to arbitrary convergent Laurent series.
Although a fairly straightforward generalization, it has not been carried out
earlier.
As our starting point we take Theorem 1, which is essentially due to Ronkin
[29], although it is stated here in slightly different terminology. The other main
results are Theorem 2, Theorem 3 which is due to Forsberg, Passare and Tsikh
[9], and Theorem 4 which was the main motivation for Gelfand, Kapranov and
Zelevinsky to introduce amoebas in [12]. The functions Φα , which are very
similar to GKZ-hypergeometric functions, appeared first in [25].
3.1
Some classical results
Theorem 1 (Ronkin [29]). Let f be a Laurent polynomial or a convergent
Laurent series. Then the following holds.
(i) The function Nf is convex.
(ii) Nf is affine linear in an open connected set E precisely if E is contained
in Acf .
(iii) If E is a connected component of Acf , then grad Nf |E is equal to the order
of E.
Proof. The convexity of Nf follows from the fact that log |f | is plurisubharmonic. Indeed, Nf (Log z) is a superposition of plurisubharmonic functions, and
22
I
Polynomial amoebas
is therefore itself plurisubharmonic. This is easily seen to be equivalent to convexity of Nf . (See also [28], Corollary 1 on p. 84.) If E ⊂ Acf then log |f | is
actually pluriharmonic in Log−1 (E), and it follows that Nf is affine linear in
E. Conversely, if Nf is affine linear in E, then Nf (Log z) is pluriharmonic in
Log−1 (E) which implies that log |f | must be pluriharmonic in Log−1 (E). But
then f (z) 6= 0 there, so E ⊂ Acf . Differentiation with respect to xj under the
integral sign in the definition (2) of Nf yields
Z
∂Nf
∂
=
log |f (z)| dη(z)
∂xj
∂xj Log−1 (0)
Z
zj ∂f /∂zj
dη(z)
= Re
f (z)
Log−1 (x)
which is precisely the real part of the second integral in (6). However, the
integral (6) is always real valued so ∂Nf /∂xj is the jth component of the order.
Hence grad Nf is equal to the order of the complement component.
An immediate consequence is the following corollaries which were proved by
other methods in [12] and [9].
Corollary 1. Every connected component of Acf is a convex open set.
Proof. It is clear that Af is closed, hence every complement component is
open. If E is a connected component of Acf of order α, then Nf (x) = c + hα, xi
in E by Theorem 1. By convexity of Nf it follows that Nf (x) ≥ c + hα, xi for
all x. Hence the set K consisting of all x such that Nf (x) = c + hα, xi is convex
and the interior of K does not intersect Af . It follows that E = int K is convex.
Corollary 2. Different components of Acf have different orders.
Proof. If E is a complement component of order α, then there exists a constant
c such that Nf (x) ≥ c + hα, xi with equality precisely in the closure of E. If
E 0 is another component of order α, and c0 the corresponding constant, then it
follows that c = c0 , and then that E = E 0 .
Hence, if Acf has a component of order α, this component is uniquely determined by α.
Definition 5. When Acf has a component of order α, this component will be
denoted Eα or Eα (f ). If Acf has no component of order α, we set Eα = ∅.
23
I
Polynomial amoebas
A simple and useful criterion for determining the order of a complement component is that a dominating term in the Laurent polynomial determines the
order of a complement component. To make this precise we introduce the following notation. If f is a convergent Laurent series whose domain of convergence
is Log−1 (Ω), and α ∈ L∗ we write
P
β6=α |fβ | exphβ, xi
mα (f ; x) =
|fα | exphα, xi
and
mα (f ) = inf mα (f ; x).
x∈Ω
Note that Ω = int{x; mα (f ; x) < +∞} for any α with fα 6= 0. The following
lemma is proved in [9].
Lemma 1. Let f (z) be a Laurent polynomial or a convergent Laurent series
and suppose that mα (f ; x) < 1 for some α ∈ L∗ and x ∈ Ω. Then x ∈ Acf and
the complement component containing x has order α.
Proof. By the triangle inequality,
|f (z)| ≥ |fα | exphα, xi −
X
β6=α
|fβ | exphβ, xi > 0
for all z ∈ Log−1 (x), so x ∈ Acf . Suppose the component containing x has order
β. By equation (6) and the argument principle,
Z
1
d log f (ex1 , . . . , ζexj , . . . , exn )
βj =
2πi |ζ|=1
!
Z
X
1
d log
fν exphν, xiζ νj
=
2πi |ζ|=1
ν
Z
1
d log(fα exphα, xiζ αj ) = αj
=
2πi |ζ|=1
so α = β.
Corollary 3. If α is a vertex of Pf , then mα (f ) = 0, hence Acf has a component
of order α.
Proof. Since α is a vertex of Pf , there exists a y ∈ LR so that hα − β, yi > 0
for all β 6= α with fβ 6= 0. Take any x ∈ Ω, where Log−1 (Ω) is the domain of
convergence of f and a positive
P number . Then mα (f ; x) < +∞, so there is
a finite set A ⊂ L∗ such that β ∈A
/ |fβ | exphβ, xi/|fα | exphα, xi < . Now, for
24
I
Polynomial amoebas
any β ∈ L∗ ∩ Pf , exphβ − α, x + tyi is decreasing as a function of t and has the
limit 0 when t → +∞. If t > 0 it follows that
mα (f ; x + ty) =
X
β∈Ar{α}
≤
X
β∈Ar{α}
|fβ | exphβ, x + tyi X |fβ | exphβ, x + tyi
+
|fα | exphα, x + tyi
|fα | exphα, x + tyi
β ∈A
/
|fβ | exphβ, x + tyi
+
|fα | exphα, x + tyi
≤ 2 for large t.
Since clearly x + ty ∈ Ω for all t ≥ 0, it follows that mα (f ) = 0 as required. If u is a convex function defined in a domain Ω ⊂ LR , then by the gradient
of u at x0 ∈ Ω we will mean the set
grad u(x0 ) = {ξ ∈ L∗R ; u(x) − u(x0 ) ≥ hξ, x − x0 i, ∀x ∈ Ω}.
(7)
When u is differentiable at x0 , grad u(x0 ) consists of a single point which is just
the usual gradient of u. By the gradient image of a set E ⊂ Ω, we will mean
the set
[
grad u(E) =
grad u(x).
(8)
x∈E
Theorem 2. If f is a Laurent series converging in Log−1 (Ω), then
grad Nf (Ω) ⊂ Pf .
If f is a Laurent polynomial, then moreover
relint Pf ⊂ grad Nf (LR ).
Proof. Let ξ ∈ grad Nf (x) for some x ∈ Ω. Assume that a ∈ L and k ∈ Z
are such that hα, ai ≥ k for all α ∈ L∗ with fα 6= 0. If z ∈ Log−1 (x), then
ζ −k f (ζ a z) is holomorphic as a function of ζ in the unit disc. Applying the
maximum principle to this function and taking ζ = e−t it follows that
sup
z∈Log−1 (x−ta)
hence
Nf (x − ta) =
Z
Log−1 (x−ta)
ekt |f (z)| ≤
sup
z∈Log−1 (x)
|f (z)|
log |f (z)| dη(z) ≤ −kt +
sup
Log −1 (x)
log |f (z)|.
It follows that
hξ, −tai ≤ Nf (x − ta) − Nf (x) ≤ −kt +
25
sup
Log−1 (x)
log |f (z)| − Nf (x).
I
Polynomial amoebas
Letting t → +∞ it follows that hξ, ai ≥ k, hence ξ ∈ Pf .
Suppose now that f is a Laurent polynomial and that ξ ∈ relint Pf . Assume
first that Pf is n-dimensional so that ξ ∈ int Pf . If α is a vertex of Pf , then
Acf has a component of order α by Corollary 3, so it follows from Theorem 1
that Nf (x) ≥ C + hα, xi for some constant C. Here we may assume that the
constant C is the same for all vertices. It follows that
Nf (x) − hξ, xi ≥ C +
max hα − ξ, xi → +∞ when x → ∞.
α∈vert Pf
Hence Nf (x) − hξ, xi has a minimum at some x0 , and then ξ ∈ grad Nf (x0 ).
The case when dim Pf < n is handled similarly. The only difference is that
Nf (x) − hξ, xi is constant along subspaces orthogonal to Pf in this case.
Theorem 3 (Forsberg, Passare, Tsikh [9]). The mapping which takes a
connected component of Acf to its order is an injection from the set of complement components to Pf ∩ L∗ .
Proof. By definition, the order of any complement component is in L∗ . That
the order of any complement component is in Pf follows from Theorem 1 and
Theorem 2. The injectivity is just Corollary 2.
The following estimates were obtained by Gelfand, Kapranov and Zelevinsky
[12] and Forsberg, Passare and Tsikh [9].
Corollary 4. The number of components of Acf is at least equal to the number
of vertices of Pf and at most equal to the number of lattice points in Pf .
There is much more to be said about the relation between the geometry of a
complement component and its order. For example, there is a strong connection
between the size of a complement component and the location of its order in
the Newton polytope. Thus, a component is bounded precisely if its order is in
the interior of the Newton polytope, while the largest components (in a certain
sense) are those which correspond to vertices of the Newton polytope. We refer
to [8], [9] and [12] for further results on these matters.
3.2
A class of almost hypergeometric functions
Definition 6. If Eα (f ) is nonempty, define
Z
Φα (f ) =
log(f (z)/z α ) dη(z),
Log−1 (x)
1
=
(2πi)n
Z
Log−1 (x)
x ∈ Eα
log(f (z)/z α ) dz1 ∧ . . . ∧ dzn
.
z1 . . . zn
26
(9)
I
Polynomial amoebas
Notice that the the function log(f (z)/z α ) has a globally defined holomorphic
branch in Log−1 (Eα ). Hence the integral (9) defines a holomorphic function in
the coefficients of f with values in C/2πiZ. The second integral shows that
the definition is independent of the choice of x ∈ Eα . In fact, the integration
may be performed over any cycle γ homologous to Log−1 (x) in LC∗ r f −1 (0)
on which log(f (z)/z α ) has a holomorphic branch. This can be used to define an
analytic continuation of Φα to all f ∈ CA whose principal A-determinant (see
[12]) is nonzero, where A ⊂ L∗ is a fixed finite set.
These functions Φα are interesting in several ways. An immediate observation is that Nf (x) ≥ hα, xi + Re Φα (f ) with equality when x ∈ Eα . It follows
that
Nf (x) ≥ max (Re Φα (f ) + hα, xi)
with equality in the closure of Acf . The maximum is taken over all α such that
Eα (f ) is nonempty. This approximation of the function Nf will be used in
section 4 to construct a polyhedral complex approximating the amoeba. We
note also that when mα (f ) < 1 we have the estimate
|Φα (f ) − log fα | ≤ − log(1 − mα (f ))
(10)
which follows immediately from the definition.
The following result was alluded to in the introduction. After giving the
precise statement we shall show that all the coefficients in a Laurent series
expansion of 1/f can be expressed in terms of the functions Φα .
Theorem 4 (Gelfand, Kapranov, Zelevinsky [12]). If f is a Laurent polynomial, then the convergent Laurent series g such that f g = 1 are in bijective
correspondence with the connected components of Acf .
Proof. If E is a complement component of Af , then 1/f is a holomorphic
function in Log−1 (E). This function is represented by a convergent Laurent
series g, and evidently f g = 1. Conversely, if g is a convergent Laurent series,
converging say in a domain Log−1 (E), with f g = 1, then the holomorphic
function defined by g in Log−1 (E) is equal to 1/f (z). It follows that f (z) 6= 0
in Log−1 (E) which means that E ⊂ Acf . If E 0 is the complement component
containing E, then g is equal to the unique Laurent series expansion of 1/f (z)
in Log−1 (E 0 ). This proves the theorem.
Theorem 5. Assume Eα (f ) is nonempty, and define cν = ∂Φα /∂fν (f ) for all
ν ∈ L∗ . Then
X
1
=
cν z −ν ,
f (z)
∗
ν∈L
the series converging in Log
−1
(Eα ).
27
I
Polynomial amoebas
Proof. Differentiation under the integral sign in (9) yields
Z
∂Φα
z ν dη(z)
=
.
∂fν
f (z)
Log−1 (x)
This is precisely the coefficient for z −ν in a Laurent series expansion of 1/f
converging in a neighborhood of Log−1 (x).
The following theorem shows that the functions Φα satisfy a system of differential equations which is very similar to a so-called A-hypergeometric system (see [11]). In fact, the only way the following equations fail to be of Ahypergeometric type is that the right-hand sides of equations (14) and (15) are
nonzero.
P
Theorem 6 ([25]). Let f (z) = ν∈A fν z ν be a general Laurent polynomial in
CA where A is a fixed finite subset of L∗ . Then the holomorphic functions Φα
have the power series expansion
Φα (f ) = log fα +
X (−kα − 1)!
Q
(−1)kα −1 f k ,
β6=α kβ !
(11)
k∈Kα
where f k =
Q
ν∈A
fνkν and
Kα = {k ∈ ZA ; kα < 0, kβ ≥ 0 if β 6= α,
X
kν = 0,
X
νkν = 0}.
(12)
ν
ν
The domain of convergence of the series is the set of all f with mα (f ) < 1.
Moreover, Φα satisfies the differential equations
X
X
(∂ u − ∂ v )Φα = 0 if
(uν − vν ) = 0 and
ν(uν − vν ) = 0
(13)
ν
ν
and
X
fν ∂ ν Φα = 1
(14)
νfν ∂ν Φα = α.
(15)
ν
X
ν
where ∂ν = ∂/∂fν and ∂ u =
Q
∂νuν when u ∈ ZA
≥0 .
Remark. Notice that the power series in (11) only involves those coefficients
of f which belong to the smallest face of Pf containing α. In particular, if α is
a vertex of Pf , then Φα (f ) = log fα .
28
I
Polynomial amoebas
Proof. Use the power series expansion of the logarithm function to write
m

X (−1)m−1 X fβ z β
 .

log(f (z)/z α ) = log fα +
m
fα z α
m≥1
β6=α
Now

m
Q kβ kβ β
X (−1)m−1 X fβ z β
X X (−1)m−1 m!
fβ z

 =
Q
α
kβ ! fαm z mα
m
fα z
m
m≥1
β6=α
m≥1 Σkβ =m
=
X
(−1)kα −1
Lα
(−kα − 1)! k Σkν ν
Q
f z
.
kβ !
Here all sums and products indexed by β are taken over β P
∈ A r {α} while
ν ranges over all of A and Lα = {k ∈ ZA ; kα < 0, kβ ≥ 0, kν = 0}. The
constant terms in this expression, considered as monomials in the z variables,
are precisely those corresponding to the set Kα , and we have proved (11).
Next we compute the domain of convergence of the power series. Let
(−kα − 1)!
(−1)kα −1 f k
θk = Q
β6=α kβ !
be the term corresponding to k ∈ Kα . P
Using Stirling’s
formula log m! =
P
m log m − m + O(log m) and the relations
kν = 0, νkν = 0 we find that
X
log |θk | = −kα log(−kα ) + kα −
(kβ log kβ − kβ )
+
X
ν
= −kα
β
kν log |fν | + O(log |kα |)
X kβ
−kα X
log
+
kν log(|fν | exphν, xi) + O(log |kα |)
−kα
kβ
ν
β
X kβ
−kα |fβ | exphβ, xi
log
+ O(log |kα |)
= −kα
−kα
kβ |fα | exphα, xi
β
for any x ∈ LR . Assume now that mα (f ) < 1 and take x so that mα (f ; x) < 1.
It follows then from Jensen’s inequality that
log |θk | = −kα
X kβ
−kα |fβ | exphβ, xi
log
+ O(log |kα |)
−kα
kβ |fα | exphα, xi
β
≤ −kα log
X |fβ | exphβ, xi
+ O(log |kα |)
|fα | exphα, xi
β
= −kα log mα (f ; x) + O(log |kα |).
29
I
Polynomial amoebas
Since the number of terms
P with a given kα increases polynomially with kα , it
follows that the series k∈Kα θk is absolutely convergent.
To prove the converse, we may assume that α is in the relative interior of
Pf . Otherwise we simply replace f with its truncation to the smallest face
containing α. This operation leaves both mα (f ) and the series (11) unchanged.
Then mα (f ; x) attains its minimal value. Take a point x where this minimum
is achieved, and note that
X
1
|fβ | exphβ, xi(β − α) = 0.
|fα | exphα, xi
P
P
Setting φβ = |fβ | exphβ, xi and φα = − φβ , this means that
φν = 0 and
P
νφν = 0, so that the vector φ belongs to the cone generated by Kα . Since Kα
is a semigroup, there is a constant C such that for any t > 0 there is a k ∈ Kα
with |k − tφ| < C. For the corresponding term θk we then have
grad mα (f ; x) =
log |θk | = −kα
X kβ
−kα |fβ | exphβ, xi
log
+ O(log t)
−kα
kβ |fα | exphα, xi
β
X kβ
−tφα φβ
= −tφα
log
+ O(log t)
kα
φβ |fα | exphα, xi
β
= −tφα log mα (f ; x) + O(log t).
If the series is to converge, these terms must remain bounded as t → +∞, which
implies that mα (f ; x) ≤ 1. Since the domain of convergence is by definition an
open set, it follows that it is defined by the inequality mα (f ) < 1.
To verify the differential equations, we differentiate under the sign of integration defining Φα . By a simple computation,
X
∂ u log(f (z)/z α ) = −
uν − 1 ! z Σuν ν (−f (z))−Σuν
P
P
which depends only on
uν and
uν ν. Also,
X
X
fν ∂ν log(f (z)/z α) =
fν z ν /f (z) = 1.
This verifies (13) and (14). Finally,
X
νj fν ∂ν log(f (z)/z α) =
X
νj fν z ν /f (z) =
zj ∂f /∂zj
.
f (z)
Comparing this to the second integral in (6) proves the relation (15).
3.3
Functorial properties
For convenience, we record here how certain changes of coordinates affect the
amoeba and the Ronkin function.
30
I
Polynomial amoebas
Theorem 7. Let T : L → M be a linear mapping between two lattices such
that T ∗ : M ∗ → L∗ is injective. Let f ∈ C[M ] be a Laurent polynomial.
−1
−1
Then TR
(Af ) = AT ∗ f , TR
(Eα (f )) = ET ∗ α (T ∗ f ), Nf (TR x) = NT ∗ f (x) and
Φα (f ) = ΦT ∗ α (T ∗ f ).
Proof. Notice that TR and TC∗ are surjective since T ∗ is injective. From
−1 −1
the relation (3) it follows that (T ∗ f )−1 (0) = TC
(f (0)). Applying Log to
∗
−1
both sides of this equality and using (4), it follows that AT ∗ f = TR
(Af ). In
−1
c
particular, TR (Eα (f )) is a connected component of AT ∗ f . That the order of
this component is T ∗ α can be seen directly from the definition. An alternative
is to use the gradient of the Ronkin function. From (5) it follows that
Z
log |f (w)| dηM (w)
Nf (TR x) =
Log−1 (TR x)
Z
log |f (TC∗ z)| dηL (z)
=
Log−1 (x)
Z
=
log |T ∗ f (z)| dηL (z)
Log−1 (x)
= NT ∗ f (x).
Finally, if x ∈ ET ∗ α (T ∗ f ), then TR x ∈ Eα (f ) and it follows that
Z
Φα (f ) =
log(f (w)/w α ) dηM (w)
−1
Log (TR x)
Z
log(f (TC∗ z)/(TC∗ z)α ) dηL (z)
=
−1
Log (x)
Z
∗
=
log(T ∗ f (z)/z T α ) dηL (z)
=
Log−1 (x)
ΦT ∗ α (T ∗ f ).
4
Patchworking amoebas
In this section, amoebas are compared to certain polyhedral subdivisions of the
space LR . It is shown that every amoeba can be approximated by a polyhedral complex of a special kind (Theorem 8) and conversely, all such polyhedral
complexes can be approximated by amoebas if rescalings are allowed (Theorem
9). The construction is reminiscent of the patchworking technique invented by
Viro for constructing real algebraic curves with prescribed topology, hence the
title. The ideas in this section are inspired by the computer generated pictures
in [8, section 5], and provide an explanation for the empirical observations made
there.
31
I
Polynomial amoebas
4.1
Dual polyhedral subdivisions
Definition 7. Let K be a polyhedron in a real vector space V (possibly all of V ).
By a polyhedral subdivision of K we will mean a finite collection Σ of nonempty
polyhedra whose union is K and satisfying the following properties.
(i) If σ, τ ∈ Σ and σ ∩ τ is nonempty, then σ ∩ τ ∈ Σ.
(ii) If σ ∈ Σ and τ ⊂ σ is a polyhedron, then τ ∈ Σ precisely if τ is a face of
σ.
If σ is a polyhedron, and τ is a face of σ, we shall denote by cone(τ, σ) =
{t(x − y); t ≥ 0, x ∈ σ, y ∈ τ } the cone of vectors pointing from τ into σ. If
C is a closed convex cone in V , its dual is defined to be the cone C ∨ = {ξ ∈
V ∗ ; hξ, xi ≤ 0, ∀x ∈ C}. It is well known that C ∨∨ = C. We shall therefore say
that two cones C1 , C2 are dual if C1∨ = C2 or equivalently C1 = C2∨ .
Definition 8. Let Σ, Σ0 be polyhedral subdivisions of polyhedra K ⊂ V, K 0 ⊂ V ∗
respectively. We shall say that Σ and Σ0 are dual if there is a bijection σ 7→ σ ∗
from Σ to Σ0 such that τ ⊂ σ if and only if σ ∗ ⊂ τ ∗ and whenever this is the
case, cone(τ, σ) and cone(σ ∗ , τ ∗ ) are dual.
Next we discuss a particular method of constructing dual polyhedral subdivisions. Let K be a polyhedron in V and let s be a piecewise linear function
on K, by which we here mean that s is the maximum of finitely many linear
functions. The Legendre transform of s,
s̃(ξ) = sup (hξ, xi − s(x))
x∈K
is also a piecewise linear function on the polyhedron K 0 consisting of all points
ξ with s̃(ξ) < +∞. Define
S(ξ, x) = s(x) + s̃(ξ) − hξ, xi.
Let Σ be the collection of all sets of the form {x ∈ K; S(ξ, x) = 0} for some
ξ ∈ K 0 and let Σ0 consist of all sets of the form {ξ ∈ K 0 ; S(ξ, x) = 0} for some
x ∈ K.
Proposition 1. With notations as in the preceding paragraph, Σ and Σ 0 are
dual polyhedral subdivisions of K and K 0 .
Proof. If E is any subset of K, define E ∗ = {ξ ∈ K 0 ; S(ξ, x) = 0, ∀x ∈ E} and
define E ∗ similarly if E ⊂ K 0 . Now it is clear that E ⊂ F ⇒ F ∗ ⊂ E ∗ , E ⊂ E ∗∗
and E ∗ = E ∗∗∗ . The main points in the proof are to show that if σ ⊂ K is
nonempty then σ ∈ Σ ⇔ σ ∗∗ = σ and that if σ ∈ Σ and τ is a nonempty subset
of σ, then cone(σ ∗ , τ ∗ ) = {ξ; hξ, xi ≤ hξ, yi, ∀x ∈ σ, y ∈ τ }. From this it follows
that x ∈ {x}∗∗ ∈ Σ for any x ∈ K, so the union of all σ ∈ Σ is equal to K and
32
I
Polynomial amoebas
that if σ, τ ∈ Σ, then (σ ∩τ )∗∗ ⊂ (σ ∗ ∪τ ∗ )∗ ⊂ σ ∗∗ ∩τ ∗∗ = σ ∩τ so that σ ∩τ ∈ Σ.
It follows also that the mapping σ 7→ σ ∗ is an inclusion reversing bijection from
Σ to Σ0 . Moreover, if σ ∈ Σ and τ is a nonempty subset of σ then it follows that
τ ∗∗ is the smallest face of σ containing τ . This shows that Σ is a polyhedral
subdivision. Finally, it is easy to see that {ξ; hξ, xi ≤ hξ, yi, ∀x ∈ σ, y ∈ τ } is
dual to cone(τ, σ) and this shows that Σ and Σ0 are dual subdivisions.
Remark. Polyhedral subdivisions which can be obtained from a piecewise
linear convex function are called coherent and play an important role in the
theory of discriminants. It is known that not all subdivisions are coherent.
Proposition 1 shows that every coherent subdivision of a polyhedron K ⊂ V is
dual to a subdivision of a polyhedron K 0 ⊂ V ∗ . Conversely, it is easy to see that
if Σ is a subdivision which is dual to some subdivision Σ0 , then Σ is coherent.
4.2
The spine of an amoeba
Let now f be a Laurent polynomial and let A be the set of all α ∈ L∗ such that
Acf has a component of order α. Take
s(x) = max (Re Φα (f ) + hα, xi)
α∈A
(16)
and take K to be all of LR . The Legendre transform s̃ is then finite precisely on
the convex hull of A which is equal to the Newton polytope of f . Let Σ and Σ0
be the polyhedral subdivisions constructed from these functions and let Sf be
the union of all polyhedra in Σ whose dimension is smaller than n. This last set
will be called the spine of Af for reasons which are obvious from the following
theorem.
Theorem 8 ([25]). Let f be a Laurent polynomial. Then Σ0 is a polyhedral
subdivision of the Newton polytope of f , and Σ is a dual subdivision of L R such
that Eα = {α}∗ ∩ Acf whenever Acf has a component of order α. Moreover, Sf
is a strong deformation retract of Af .
Remark. Other choices of the function s may also produce a polyhedral subdivision compatible with the amoeba in the sense of Theorem 8. For example,
the proof of Theorem 12 shows that, under the assumptions of that Theorem,
the constants Re Φα (f ) in (16) may be replaced by log |fα |.
Proof. Since Re Φα (f ) + hα, xi ≤ Nf (x) with equality precisely in the closure
of Eα and s̃(α) = − Re Φα (f ) it follows immediately from the definition that
Eα = {α}∗ ∩Acf . Now let σ ∗ be a polyhedron in Sf where σ ∈ Σ0 with dim σ ≥ 1.
Take two points α, β ∈ σ ∩A. Then it follows that σ ∗ ∩Acf ⊂ Eα ∩Eβ = ∅, which
proves that Sf ⊂ Af . Since every connected component of the complement of Sf
is a convex polyhedron {α}∗ which contains exactly one nonempty component
Eα of Acf it follows easily that Sf is a strong deformation retract of Af .
33
I
Polynomial amoebas
In most cases it is of course much easier to compute Φα (f ) for all α ∈ A
than to compute the exact shape of the amoeba. Hence, the spine provides
an approximation to the amoeba which is much more convenient to compute
explicitly. It should be noted, however, that the spine cannot be used to find
the number of complement components of the amoeba, since the set A of orders
of the complement components is needed in the construction of the spine.
We now reverse the operation and construct amoebas approximating a prescribed polyhedral subdivision. Let A be a finite subset of L∗ and cα be an
arbitrary real number for every α ∈ A. Let
s(x) = max (cα + hα, xi),
α∈A
let Σ and Σ0 be the polyhedral subdivision of LR and conv A determined by s
and let S be the union of all polyhedra in Σ of dimension smaller than n. Also,
let A0 ⊂ A be the set of vertices of the subdivision Σ0 . Let f t ∈ CA be a family
of Laurent polynomials such that log |fαt | = tcα for all α ∈ A. If dist(E, F )
denotes the Hausdorff distance between two sets E, F in LR (with respect to
any norm), then we have the following result.
Theorem 9. With notations as in the preceding paragraph, dist(t−1 Af t , S) → 0
and dist(t−1 Eα (f t ), {α}∗ ) → 0 for all α ∈ A0 when t → +∞.
Remark. Note that the complements of these amoebas will be dominated
by the components whose orders are vertices of the dual subdivision Σ0 of the
Newton polytope. However, we are not asserting that these will be the only components in the complement of the amoeba. In general, there will be components
which have other orders as well.
Proof. For every α ∈ A0 and δ > 0, let
Fαδ = {x; cα + hα, xi − δ ≥ cβ + hβ, xi, ∀β 6= α}
and
Gδα = {x; cα + hα, xi + δ ≥ cβ + hβ, xi, ∀β 6= α}.
Then Fαδ ⊂ {α}∗ ⊂ Gδα and Fαδ and Gδα converge to {α}∗ when δ → 0. If N is
the cardinality of A and x ∈ Fαδ , then it is easy to see that mα (f t ; tx) ≤ N e−tδ .
It follows from Lemma 1 that Fαδ ⊂ t−1 Eα (f t ) for sufficiently large t. Moreover,
it follows that mα (f t ) → 0 when t → +∞, hence by (10), t−1 Re Φα (f t ) → cα
for all α ∈ A0 . Let ctα = t−1 Re Φα (f t ) and st (x) = maxα∈A0 (ctα + hα, xi). For
sufficiently large t it follows that t−1 Eα (f t ) ⊂ {x; ctα + hα, xi = st (x)} ⊂ Gδα .
Hence dist(t−1 Eα (f t ), {α}∗ ) → 0. Since t−1 Af t ⊂ Rn r ∪Fαδ and every line
segment with endpoints in Fαδ and Fβδ where α 6= β intersects t−1 Af t when t is
sufficiently large, it follows that dist(t−1 Af t , S) → 0.
34
I
Polynomial amoebas
5
The complement components of an amoeba
Gelfand, Kapranov and Zelevinsky posed in [12] the following problem: Given
a polynomial f , find all connected components of Acf . At that time, the order
of a complement component had not been defined. In view of this concept
and Theorem 3, it is natural to reformulate the problem as follows: Given a
polynomial f , find all α ∈ Pf ∩ L∗ such that Acf has a component of order α.
The present section is concerned with variations of this problem. We give
some partial answers to certain questions relating to the original problem. In
this context, we should also mention the work of Sadykov [31] on the amoebas
of certain special functions.
We introduce the sets UαA of all Laurent polynomials f ∈ CA such that
c
Af has a component of order α. The fact that these sets are semialgebraic
means that there exists, at lest in principle, a procedure for determining whether
Acf , for a given f , has a component of order α. Next we pose the following
problem: When is UαA nonempty? Theorem 11 gives one necessary and one
sufficient condition, although the gap between them is, for most sets A, very
large. Theorem 12 gives the complete answer for certain simple sets A, which
include cases where it is nontrivial to compute amoebas explicitly.
In the previous section it was shown that by choosing certain coefficients of
a polynomial f appropriately, the corresponding complement components can
be made very large. In the opposite direction, we show now that the coefficients
may be chosen in such a way that certain prescribed complement components
vanish altogether. This shows that the set of lattice points occuring as orders
of complement components of Acf , where the Newton polytope Pf is given, is
subject only to those restrictions imposed by Corollary 3 and Theorem 3. In
particular, the estimates on the number of complement components given in
Corollary 4 are sharp.
Finally, we prove a statement concerning the topology of the sets UαA .
Definition 9. If A ⊂ L∗ and α ∈ L∗ , let UαA denote the set of all f ∈ CA such
that Eα (f ) 6= ∅.
Theorem 10. All the sets UαA are open and semialgebraic.
Proof. In the product space CA × LC∗ , consider the algebraic surface V =
{(f, z); f (z) = 0}. By the Tarski-Seidenberg theorem, V is mapped onto a
semialgebraic set by the mapping φ : (f, z) 7→ (f, (|z1 |2 , . . . , |zn |2 )). Now the set
{(f, x) ∈ CA ×Rn>0 ; 21 Log x ∈ Eα (f )} consists of certain connected components
of CA × Rn>0 r φ(V ), and hence is semialgebraic. Therefore its projection on
CA , which is precisely UαA , is also semialgebraic.
Let us next turn to the following problem. For which α is UαA nonempty? Let
aff A denote the affine lattice generated by A. Then we have
35
I
Polynomial amoebas
Theorem 11. A necessary condition for UαA to be nonempty is that α ∈
conv A ∩ aff A.
A sufficient condition for UαA to be nonempty is that there exists a line l
such that α ∈ conv(A ∩ l) ∩ aff(A ∩ l).
Proof. If f ∈ UαA it follows from Theorem 3 that α ∈ Pf ⊂ conv A. Moreover,
we may assume that aff A contains 0, and then it follows from Theorem 7 with
T ∗ the inclusion mapping aff A → L∗ that α ∈ aff A. This proves the first part.
Suppose now that l is a line with α ∈ aff(A ∩ l) ∩ conv(A ∩ l). Without
loss of generality we may assume that A ⊂ l. In fact, we may assume that
A ⊂ Z with aff A = Z and that the largest element in A is N and the smallest
elementP0. Consider polynomials of the form f (z) = z N − 1 + eiθ g(z) where
g(z) =
gα z α and the sum is taken over A r {0, N }. Then the zeros of f are
given by ω j (1 − eiθ g(ω j )/N ) + o(), j = 0, . . . , N − 1 where ω = e2πi/N . For
generic choices of the coefficients gα , all the numbers g(ω j ) are distinct, hence
for suitable θ, Re(eiθ g(ω j )) are all distinct. It follows that for sufficiently small
, all the zeros of f have different absolute values, so Acf has components of all
orders between 0 and N .
Theorem 12. Suppose that A ⊂ L∗ has no more than 2n points and that no
k + 2 of these lie in an affine k-dimensional subspace for k = 1, . . . , n − 1.
If f ∈ CA has a component of order α, then α ∈ A. In other words, UαA is
nonempty if and only if α ∈ A.
Proof. Let f ∈ CA and take a point x ∈ Acf . Without loss of generality,
P2n
assume that x = 0. Write f (z) = j=1 cj z αj with |c1 | ≥ |c2 | ≥ . . . ≥ |c2n |.
Suppose the complement component containing x has order α. We will show
that α = α1 . There is no loss of generality in assuming that α1 = 0 and that
c1 ≥ 0, otherwise we divide f by c1 z α1 . For each k = 2, . . . , n + 1 choose
ak ∈ L orthogonal to α2 , . . . , αk−1 , αk+1 , . . . , αn+1 (ak is uniquely determined
α
up to scalar multiplication) and a point zk ∈ Log−1 (0) such that cj zk j ≥ 0 for
j = 2, . . . , n + 1, j 6= k. Since Re f (ζ ak zk ) ≥ |c1 | + . . . + |ck−1 | + |ck+1 | + . . . +
|cn+1 | − |ck | − |cn+2 | − . . . − |c2n | ≥ 0 whenever |ζ| = 1 it follows that
Z
1
d log f (ζ ak zk ) = 0.
hα, ak i =
2πi |ζ|=1
Since a2 , . . . , an+1 is a basis for LR it follows that α = 0 = α1 as required.
Theorem 13. Let f ∈ CA and let B ⊂ A. Then there exists a polynomial
g ∈ CB such that Eα (f + g) = ∅ for all α ∈ B.
Proof. We may assume that Pf = conv A. Suppose B contains some vertices
of Pf and let B0 = B ∩ vert Pf . We begin by taking g ∈ CB0 so that gα = −fα
36
I
Polynomial amoebas
for all α ∈ B0 . Then Pf +g does not contain any point of B0 . If B r B0 contains
vertices of Pf +g the procedure can be repeated. Hence we can assume from
the beginning that B does not contain any vertices of Pf . This implies that
mα (f + g) is continuous as a function of g ∈ CB (with values in R ∪ {+∞}) for
all α. Assume also that fα = 0 for all α ∈ B. Let
Xα = {g ∈ CB ; Eα (f + g) = ∅}
and
Yα = {g ∈ CB ; mα (f + g) > 1/2}
for all α ∈ B. By Lemma 1, Yα is an open neighborhood of Xα . Let Vα be any
open neighborhood of Xα in Yα and let φα be smooth functions defined in CB
which satisfy φα (g) = exp(−Φα (f + g)) outside Vα . Then φα is holomorphic
outside Vα and by (10)
|gα φα (g) − 1| ≤ Cmα (f + g),
g∈
/ Yα
where C is a constant (we may take C = 2e). Consider the differential form
^
^
¯ α ∧ dgα ) =
ω=
(∂φ
(dφα ∧ dgα ).
α∈B
α∈B
Notice that ω has its support in ∩Vα . Hence, if it can be shown that ω 6= 0, it
will follow that ∩Vα 6= ∅. Since Vα were arbitrarily small neighborhoods of Xα
and Xα are closed this implies that ∩Xα 6= ∅, which is precisely what we want
to prove.
To prove that ω 6= 0, we will evaluate its integral over CB . Order the
elements in B = {α1 , . . . , αk } and let dg = dgα1 ∧. . .∧dgαk and φ = φα1 . . . φαk .
Lemma 2. Let D ⊂ CB be a polydisc centered at the origin whose distinguished
boundary ∂ 0 D does not meet ∪Yα , and let ψ be holomorphic in a neighborhood
of D. Then
Z
Z
ψω =
ψφ dg.
∂0D
D
Proof. Use induction on the number of elements in B. If k = 1 this is a
simpleVapplication of Stokes’ theorem. In the general case let B 0 = B r {α1 },
¯ α ∧ dgα ), and write D = D1 × D0 with D1 ⊂ C and D0 ⊂ CB 0 .
ω 0 = α∈B 0 (∂φ
Since mα is decreasing as a function of |gα | and increasing as a function of |gβ |
for any β 6= α, it follows that ∂D1 × D0 does not intersect Yα1 and D1 × ∂D0
does not intersect ∩α∈B 0 Yα . Hence φα1 is holomorphic on ∂D1 × D0 and ω 0 = 0
on D1 × ∂D0 . By Stokes’ theorem and the inductive hypothesis it follows that
Z
Z
Z
Z
ψφα1 ω 0
dgα1
ψω =
ψφα1 dgα1 ∧ ω 0 =
D0
∂D1
D
∂D
Z
Z
Z
0
=
dgα1
ψφ dg =
ψφ dg.
∂D1
∂ 0 D0
∂0D
37
I
Polynomial amoebas
Returning to the proof of the theorem, let s : LR → R be a strictly concave
function which is positive on A. Let D(t) ⊂ CB be the polydisc {g ∈ CB ; |gα | <
ets(α) }. For large t, we then have ts(β) > log |fβ |. For every α ∈ A there is
some xα ∈ LR such that s(α) + hα, xα i > s(β) + hβ, xα i for all β 6= α. It follows
that mα (f + g) ≤ mα (f + g; txα ) → 0 uniformly for g ∈ ∂ 0 D(t) as t → +∞ for
all α ∈ B. In particular, the hypothesis of Lemma 2 is satisfied for large t and
ψ ≡ 1. It follows that
Z
Z
Z
dg
φ dg = lim
ω = lim
= (2πi)k ,
t→+∞
t→+∞
g
.
.
.
g
0
0
A
αk
∂ D(t) α1
∂ D(t)
C
so that certainly ω 6= 0. This completes the proof.
Corollary 5 ([30]). Let f ∈ CA and let B, C be disjoint subsets of A. Then
there exists a polynomial g ∈ CB∪C such that Eα (f + g) 6= ∅ for all α ∈ B and
Eα (f + g) = ∅ for all α ∈ C.
Proof. Assume that fα = 0 for all α ∈ B ∪ C. Let N be the number of
elements in A and take a strictly concave function s which is positive on A. For
every α ∈ A, take a point xα ∈ LR such that s(α) + hα, xα i > s(β) + hβ, xα i
for all β ∈ A r {α}. After multiplying s by a large constant and modifying xα
accordingly, we may assume that
s(α) + hα, xα i > s(β) + hβxα i + log N
(17)
and that s(α) > log |fα | for all α ∈ A. Now take a polynomial g 1 ∈ CB such
that log |gβ | = s(β) for all β ∈ B. By Theorem 13 there is a polynomial g 2 ∈ CC
such that Eγ (f + g 1 + g 2 ) = ∅ for every γ ∈ C. Write h = f + g 1 + g 2 and
take an α ∈ A with log |hα | − s(α) maximal. Then it follows from (17) that
mα (h, xα ) < 1, so that Eα (h) 6= ∅. Hence α ∈
/ C, but then by the construction
of s and g 1 , log |hα | − s(α) ≤ 0. Since log |hβ | = s(β) for all β ∈ B, it follows
that Eβ (h) 6= ∅ for all such β. Therefore g = g 1 +g 2 has the required properties.
Corollary 6. If P is a lattice polytope in L∗R and A is a subset of P ∩ L∗
containing all vertices of P , then there is a Laurent polynomial f , with P f = P
such that Eα (f ) is nonempty precisely if α ∈ A. In particular, the estimates in
Corollary 4 are sharp.
One might also try to prove statements about the topology of the sets UαA .
Since UαA is invariant under multiplication by nonzero scalars, it is reasonable to
consider the projective sets ŨαA = {[f ]; f ∈ UαA } ⊂ PCA . The following result
is most conveniently stated in the projective setting.
38
I
Polynomial amoebas
Theorem 14 ([30]). The intersection of the complement of ŨαA with any complex line in PCA is nonempty and connected.
Remark. This is indeed a very special statement. However, it is just about
all that can be said about the topology of the intersection of a general ŨαA with
a general complex line. For instance, the intersection of ŨαA with a complex
line can have any number of connected components, as Example 5 in section 9
shows. It seems to be an open question whether UαA always is connected.
Proof. For any f ∈ CA , let
uα (f ) = inf (Nf (x) − hα, xi)
x∈LR
= Re Φα (f )
if f ∈ UαA .
These are plurisubharmonic as a consequence of Kiselman’s minimum principle
(see [19]) and pluriharmonic in UαA .
Let l be a complex line in PCA . If l is contained in the complement of
A
Ũα there is nothing to prove. Otherwise, take two points on l represented by
Laurent polynomials f and g with g ∈ UαA . Let K = {t ∈ C; f + tg ∈
/ UαA } and
write Φ(t) = Φα (f + tg) for t ∈ C r K and u(t) = uα (f + tg). We want to show
that K is connected. Suppose K 0 is a closed and relatively open subset of K.
Let ω be a bounded open set with smooth boundary such that K 0 = K ∩ ω and
define
Z
1
d Im Φ.
N (K 0 ) =
2π ∂ω
Then N (K 0 ) is an integer, and if K 0 , K 00 are disjoint, N (K 0 ∪ K 00 ) = N (K 0 ) +
N (K 00 ). We intend to show that N (K 0 ) ≥ 1 whenever K 0 is nonempty and that
N (K) = 1. From this it follows that K is nonempty and connected.
First, since u is subharmonic
Z
Z
Z
1
d Im Φ =
dc u =
ddc u ≥ 0
2π ∂ω
∂ω
ω
¯
where dc = (∂ − ∂)/2πi.
Equality occurs if and only if u is harmonic in ω.
In this case, Φ can be continued analytically across ω. For the same reason,
Φα (f + tg + sz ν ) has an analytic continuation to t ∈ ω for all ν ∈ L∗ and
sufficiently small s. Set
∂
Φα (f + tg + sz ν )s=0
∂s
P
and consider the Laurent series ν∈L∗ cν (t)z −ν . When t ∈ ∂ω it follows from
Theorem 5 that this is a Laurent series expansion of 1/(f + tg) which converges
in Log−1 (Eα (f + tg)). It follows from the maximum principle that the series is
cν (t) =
39
I
Polynomial amoebas
convergent for all t ∈ ω. This implies that Acf +tg has a component of order α
for all t ∈ ω, which means that K 0 = ∅.
Hence N (K 0 ) ≥ 1 if K 0 is nonempty. From the fact that
Φ(t) = log t + Φα (f /t + g) = log t + Φα (g) + O(|t|−1 )
looks asymptotically like log t when t → ∞ it follows that N (K) = 1. This
completes the proof.
6
Convexity of the Ronkin function
According to Theorem 1, the function Nf is affine linear precisely in the complement of the amoeba of f . This section is concerned with the question of how
much Nf deviates from being linear at points in the amoeba.
If u is a smooth convex function, its Hessian
∂2u
Hess(u) =
∂xj ∂xk
is a positive definite matrix which in a certain sense measures how convex u is.
The trace and determinant of the Hessian matrix are respectively the Laplace
and the Monge-Ampère operator. We will see that the Monge-Ampère operator
has useful properties for studying convexity of the function Nf .
After a brief discussion of the Monge-Ampère operator we define the MongeAmpère measure µf of the function Nf . One motivation for choosing to work
with the Monge-Ampère operator is Theorem 15, which relates the total mass
of µf to the Newton polytope Pf . Next we relate the measure µf to local
properties of the hypersurface f −1 (0). Finally, the Monge-Ampère measure is
used to derive an estimate on the area of amoebas in the two dimensional case.
It turns out, that the amoebas with maximal area, for a give Newton polytope,
correspond to polynomials defining so-called Harnack curves which arise in real
algebraic geometry.
6.1
The Monge-Ampère operator
Let Ω be a domain in Rn . The Monge-Ampère operator is defined on smooth
convex functions in Ω as the determinant of the Hessian matrix. More precisely,
M u = det Hess(u) · λ
(18)
is called the Monge-Ampère measure of u, where λ denotes Lebesgue measure.
The reason for defining M u as a measure is that the definition can then be
extended to all convex functions without any requirements on smoothness. For
an arbitrary convex function u and a Borel set E,
M u(E) = λ(grad u(E))
40
I
Polynomial amoebas
where grad u(E) is defined by (7) and (8). It can be shown, although it is not
entirely obvious, that this defines a positive Borel measure M u for any convex
function u. Moreover, M is a continuous operator from the space of convex
functions with the topology of uniform convergence on compact sets, to the
space of measures with the weak topology. A good reference for the MongeAmpère operator is [27].
When n > 1, the Monge-Ampère operator is not linear. However, it can
be turned into a multilinear operator, taking n convex functions as arguments.
The construction is described in Proposition 2. This multilinear operator will
be called the mixed Monge-Ampère operator in analogy with the term mixed
volume in the theory of convex bodies.
Let A be a real n × n matrix whose entries are considered as indeterminates.
The determinant of A is then a homogeneous polynomial of degree n. From this
it follows that
n
det(A1 , . . . , An ) =
1 X
n!
X
(−1)n−k det(Aj1 + . . . + Ajk ).
(19)
k=1 1≤j1 <...<jk ≤n
defines a symmetric multilinear form on the space of all n × n matrices, with
the property that det A = det(A, . . . , A). It is known that when A1 , . . . , An are
positive definite (in particular, symmetric) det(A1 , . . . , An ) is positive. In this
case it actually holds that (see [1])
det(A1 , . . . , An ) ≥ (det A1 . . . det An )1/n .
(20)
From this it is easy to deduce
e taking n convex functions as
Proposition 2. There exists a unique operator M
e
arguments with the properties that M(u1 , . . . , un ) is a positive measure depending
e
multilinearly and symmetrically on u1 , . . . , un and M(u,
. . . , u) = M u for every
u.
Proof. If there exists an operator with the required properties, it is easy to
show that it must satisfy
n
X
e 1 , . . . , un ) = 1
M(u
n!
X
(−1)n−k M(uj1 + . . . + ujk ).
(21)
k=1 1≤j1 <...<jk ≤n
e by the formula
Hence uniqueness is established. To prove existence, we define M
(21), and prove that it has the required properties.
If u1 , . . . , un are two times differentiable and we set Aj = Hess(uj ), it follows
e 1 , . . . , un ) = det(A1 , . . . , An ) · λ is a positive measure depending multhat M(u
e
tilinearly and symmetrically on u1 , . . . , un and that M(u,
. . . , u) = M(u). The
general case follows by approximating uj with smooth functions and passing to
the limit.
41
I
Polynomial amoebas
When u1 , . . . , un are convex functions there is in general no geometric ine 1 , . . . , un )(E) in terms of the gradient images grad uj (E).
terpretation of M(u
However, if E = Rn we have the following
Proposition 3. For a convex function u defined in Rn , let Ku be the closure of
grad u(Rn ). Then Ku is convex, and if it is also compact, then the total mass
of M u equals Vol(Ku ). If Ku1 , . . . , Kun are all compact, then the total mass of
e 1 , . . . , un ) is equal to the mixed volume Vol(Ku1 , . . . , Kun ).
M(u
Proof. Assume without loss of generality that u(0) = 0. For t ≥ 1 define
ut (x) = u(tx)/t. Then ut is an increasing family of convex functions and
u∞ (x) = sup ut (x) is a positively homogeneous convex function with values
in R ∪ {+∞}. Let Ku0 = {ξ; hξ, xi ≤ u∞ (x), ∀x ∈ Rn }. It is clear that Ku0 is a
closed convex set. If ξ ∈ grad u(Rn ), then u(x) ≥ c + hξ, xi for some constant c,
hence ut (x) ≥ c/t + hξ, xi and u∞ (x) ≥ hξ, xi. It follows that grad u(Rn ) ⊂ Ku0 .
On the other hand, if ξ ∈ int Ku0 , then u∞ (x) − hξ, xi ≥ c|x| for some constant
c > 0. It follows that for sufficiently large t, ut (x) − hξ, xi > 0 when x ∈ ∂B,
where B is the unit ball, so ξ ∈ grad ut (B) ⊂ grad u(Rn ). Similarly, if ξ ∈
relint Ku0 it follows that ξ ∈ grad u(Rn ). Hence relint Ku0 ⊂ grad u(Rn ) ⊂ Ku0 ,
and it follows that Ku = Ku0 is convex. It follows also that the total mass of
M u is equal to Vol(Ku ). Moreover, it is clear that (u + v)∞ = u∞ + v∞ for
all convex functions u and v, and if Ku and Kv are compact this implies that
e 1 , . . . , un ) and Vol(Ku1 , . . . , Kun ) both depend
Ku+v = Ku + Kv . Hence M(u
multilinearly on u1 , . . . , un . Since we have already shown that the total mass
e
of M(u,
. . . , u) = M u is equal to Vol(Ku , . . . , Ku ) = Vol(Ku ) it follows that the
e 1 , . . . , un ) is equal to Vol(Ku1 , . . . , Kun ).
total mass of M(u
There is also a complex version of the Monge-Ampère operator. If U is a
smooth plurisubharmonic function defined in Cn , then integration of the dif¯
ferential form (ddc U )n (where dc = (∂ − ∂)/2πi)
defines the Monge-Ampère
measure of U . The multilinear version of the complex Monge-Ampère measure
is defined by integrating the form ddc U1 ∧ . . . ∧ ddc Un where U1 , . . . , Un are
smooth plurisubharmonic functions. The complex Monge-Ampère operator can
be extended to a continuous operator from the space of continuous plurisubharmonic functions with the topology of uniform convergence on compact sets, to
the space of measures with the weak topology (see [2]).
If u is a convex function defined in Rn , we may define a plurisubharmonic
function in Cn∗ by
U (z) = u(Log z).
(22)
Similarly, if U is a plurisubharmonic function defined in Cn∗ , then
Z
u(x) =
U (z) dη(z)
Log −1 (x)
42
(23)
I
Polynomial amoebas
is a convex function defined in Rn . The real and complex Monge-Ampère operators are related by the following properties.
Proposition 4. If u1 , . . . , un are convex functions and Uj (z) are defined by
(22), then
Z
Z
e 1 , . . . , un ) = 1
M(u
ddc U1 ∧ . . . ∧ ddc Un .
(24)
n!
−1
E
Log (E)
for any Borel set E. If U1 , . . . , Un are continuous plurisubharmonic functions
and uj (x) are defined by (23), then
Z
Z
Z
e 1 , . . . , un ) = 1
ddc U1 (t(1) z) ∧ . . . ∧ ddc Un (t(n) z)dη 0 (t)
M(u
n! Tn2 Log−1 (E)
E
(25)
2
where Tn denotes the real n2 -dimensional torus
(k)
(k)
(k)
{t = (tj ); tj
∈ C, |tj | = 1, j, k = 1, . . . , n}
(k)
(k)
with the normalized Haar measure η 0 , and t(k) = (t1 , . . . , tn ) acts on Cn∗ by
componentwise multiplication.
Proof. We first prove (24) in the case where u1 = . . . = un = u is a smooth
¯
function (see also [26]). Let U (z) = u(Log z). Since ddc U = i∂ ∂U/π
we have
n
2
i
∂ U
(ddc U )n = n!
dz1 ∧ dz̄1 ∧ . . . ∧ dzn ∧ dz̄n .
det
π
∂zj ∂ z̄k
Moreover, since
∂2U
1
∂2u
=
∂zj ∂ z̄k
4zj z̄k ∂xj ∂xk
it follows that
det
∂2U
∂zj ∂ z̄k
=
1
det
n
2
4 |z1 | . . . |zn |2
∂2u
∂xj ∂xk
.
Writing zj = exp(xj + iyj ), we have dzj ∧ dz̄j = −2i|zj |2 dxj ∧ dyj and it follows
that
n!
∂2u
(ddc U )n =
det
dx1 ∧ dy1 ∧ . . . ∧ dxn ∧ dyn .
(2π)n
∂xj ∂xk
Hence
Z
(ddc U )n
Log−1 (E)
n!
=
(2π)n
Z
det
E
∂2u
∂xj ∂xk
dx1 ∧ . . . ∧ dxn
Z
0<yj <2π
dy1 ∧ . . . ∧ dyn
= n!
43
Z
M u.
E
I
Polynomial amoebas
The proof of (24) is complete in this special case. The case with arbitrary
convex u is obtained by a limiting procedure. Recall that both the real and
the complex Monge-Ampère operators can be extended in a continuous way to
all convex and continuous plurisubharmonic functions respectively. Finally, the
result is extended to arbitrary convex functions u1 , . . . , un by observing that
both sides of (24) depend multilinearly and symmetrically on u1 , . . . , un .
To prove (25), let Uj be continuous plurisubharmonic functions and let uj
be defined by (23). Also, write Ũj (z) = uj (Log z). Then it follows by reversing
the order of integration that
Z
Z
ddc U1 (t(1) z) ∧ . . . ∧ ddc Un (t(n) z)dη 0 (t)
Tn2 Log−1 (E)
Z
Z
e 1 , . . . , un ).
M(u
ddc Ũ1 ∧ . . . ∧ ddc Ũn = n!
=
Log−1 (E)
E
6.2
Monge-Ampère measures on amoebas
Throughout this section, Ω denotes a convex domain in Rn and f, f1 , f2 , . . .
denote holomorphic functions, all defined in Log−1 (Ω) unless specified otherwise.
Definition 10 ([25]). Define µf to be the Monge-Ampère measure M Nf , and
e f1 , . . . , Nfn ).
µf1 ,...,fn to be the mixed Monge-Ampère measure M(N
Theorem 15. The measure µf has its support in Af , and µf1 ,...,fn has its
support in Af1 ∩ . . . ∩ Afn . If f, f1 , . . . , fn are Laurent polynomials, then the
total mass of µf equals Vol(Pf ) and the total mass of µf1 ,...,fn equals the mixed
volume Vol(Pf1 , . . . , Pfn ).
Proof. Since Nf is affine linear outside Af it follows from the definition of
the Monge-Ampère operator that supp µf ⊂ Af and that supp µf1 ,...,fn ⊂ Af1 ∩
. . . ∩ Afn . The statement about the total mass follows directly from Proposition
3 and Theorem 2.
Theorem 16 ([25]). If E is a Borel set in Ω, then µf1 ,...,fn equals the average
number of solutions in Log−1 (E) to the system of equations
(j)
fj (t1 z1 , . . . , t(j)
n zn ) = 0
(j)
j = 1, . . . , n
(26)
2
as t = (tk ) ranges over the torus Tn .
In the proof we will need the following lemma.
Lemma 3. Let V be an analytic set in a neighborhood of a compact set D 1 ×
D2 ⊂ Ck × Cm . Then there exists a constant C such that V ∩ {z} × D2 either
has positive dimension or has at most C points for every z ∈ D1 .
44
I
Polynomial amoebas
Proof. Take any point, say (0, 0) in D1 ×D2 . It suffices to prove the statement
with D1 and D2 replaced by arbitrarily small neighborhoods of (0, 0). Use
induction on m. If dim(V ∩ {0} × D2 ) < m then V ∩ D1 × D2 can be properly
projected to D1 × D20 ⊂ Ck × Cm−1 , and this way the problem is reduced
to a smaller m. So we may assume that the statement is already proved when
dim(V ∩{0}×D2 ) < m. Assume therefore that V ⊃ {0}×D2 and let fj (z, w) =
P
fj,α (z)wα be defining functions for V near the origin. Consider the ideal
generated by all fj,α in the ring of germs of holomorphic functions, and let
gP
1 , . . . , gr be a finite subset of the fj,α generating the same ideal. Write fj,α =
φj,α,l gl . Since for every l there is some j, α with gl = fj,α we may assume that
the corresponding φj,α,l ≡ 1. Now consider
set W in Ck × Cr × Cm
P the analytic
α
defined by the functions hj (z, ζ, w) =
φj,α,l (z)ζl w . If ζ 6= 0, then some ζl 6=
0 and if j, α are such that φj,α,l ≡ 1, it follows that the function w 7→ hj (z, ζ, w)
is not identically 0 for any z. Hence dim(W ∩ {z} × {ζ} × D2 ) < m whenever
ζ 6= 0. By the inductive hypothesis we may then assume that W ∩{z}×{ζ}×D2
either has positive dimension or has no more than C points for all z ∈ D1 and
all ζ with |ζ| = 1. But since hj (z, tζ, w) = thj (z, ζ, w), the same estimate holds
for all ζ 6= 0. By taking ζl = gl (z), it follows that V ∩ {z} × D2 either has
positive dimension or has no more than C points for any z ∈ D1 .
Proof of Theorem 16. If Uj are smooth plurisubharmonic functions which
converge to log |fj |, then uj defined by (23) converge to Nfj . By the continuity
of the real Monge-Ampère operator this implies that M̃ (u1 , . . . , un ) converges
to M̃ (Nf1 , . . . , Nfn ) in the weak topology. Also ddc U1 (t(1) z) ∧ . . . ∧ ddc Un (t(n) z)
converges weakly to the sum of point masses at the solutions of f1 (t(1) z) = . . . =
fn (t(n) z) = 0. Hence the theorem follows by using (25) and passing to the limit
if we only show that
Z
ddc U1 (t(1) z) ∧ . . . ∧ ddc Un (t(n) z)
Log−1 (E)
2
remains uniformly bounded as Uj → log |fj | for almost all t ∈ Tn . Here we
may assume that E is compact and that Uj is of the form Uj = ψ(log |fj |) where
ψ is a convex function, constant near −∞, which will converge to the identity
function. Let ft (z) = (f1 (t(1) z), . . . , fn (t(n) z)). Then ft (z) is a holomorphic
function in z and t defined for z in a neighborhood of Log−1 (E) and t in a
2
complex neighborhood of Tn . By Lemma 3 there exists a constant C such that
the number of solutions z in Log−1 (E) to the equation ft (z) = w is bounded
2
above by C for almost all t ∈ Tn and w ∈ Cn . Since ω = ddc ψ(log |w1 |) ∧ . . . ∧
ddc ψ(log |wn |) induces a positive measure on Cn with total mass 1, it follows
that
Z
ft∗ ω ≤ C
0≤
Log −1 (E)
for almost all t, and this completes the proof.
45
I
Polynomial amoebas
Theorem 16 can be thought of as a local analog of Bernstein’s theorem relating
the number of solutions to a system of polynomial equations to the mixed volume
of the Newton polytopes. In fact, Bernstein’s result can rather easily be derived
from Theorem 16.
Corollary 7 (Bernstein’s theorem [3]). Let P1 , . . . , Pn be integer polytopes
in Rn and let f1 , . . . , fn be generic Laurent polynomials subject to the restriction
that Pfj = Pj . Then the number of solutions in Cn∗ of the system of equations
f1 (z) = . . . = fn (z) = 0 is equal to n! Vol(P1 , . . . , Pn ).
Proof. We assume it is known that the number of solutions is equal to some
constant N when (f1 , . . . , fn ) is outside some subvariety in the space of n-tuples
of polynomials with Newton polytopes P1 , . . . , Pn . If f1 (z) = . . . = fn (z) = 0
has the generic number of solutions, then the same is true for f1 (t(1) z) = . . . =
2
fn (t(n) z) = 0 for almost all t ∈ Tn . It follows from Theorem 15 and Theorem
16 that N = n! Vol(P1 , . . . , Pn ).
Theorem 17. If u1 , . . . , un−1 are convex functions defined in Ω and Uj (z) =
uj (Log z), then
Z
Z
e 1 , . . . , un−1 , Nf ) =
n!
ddc U1 ∧ . . . ∧ ddc Un−1
M(u
Log−1 (E)∩f −1 (0)
E
for any Borel set E ⊂ Ω.
Proof. Since ddc log |f | is equal to the current of integration along f −1 (0) it
follows from (24) that
Z
Z
e 1 , . . . , un−1 , Nf ) =
n!
M(u
ddc U1 ∧ . . . ∧ ddc Un−1 ∧ ddc Nf (Log z)
−1
E
Log (E)
Z
ddc U1 ∧ . . . ∧ ddc Un−1 ∧ ddc log |f |
=
−1
Log (E)
Z
=
ddc U1 ∧ . . . ∧ ddc Un−1 .
Log−1 (E)∩f −1 (0)
Corollary 8 ([25]). If ∆ denotes the Laplace operator and E is any Borel set
in Ω, then
Z
Z
(n − 1)!
∆Nf =
ω n−1
E
Log−1 (E)∩f −1 (0)
where ω = (|z1 |−2 dz̄1 ∧ dz1 + . . . + |zn |−2 dz̄n ∧ dzn )/2πi. Hence (n − 1)!∆Nf is
the direct image of ω|f −1 (0) (considered as a measure) under the mapping Log.
46
I
Polynomial amoebas
Proof. This follows immediately from Theorem 17 since
and ω = ddc | Log z|2 .
2
e
∆Nf = n M(|x|
, . . . , |x|2 , Nf )
Finally, we shall give a formula for µf in the two-dimensional case which has
some interesting consequences. Let Ω be a convex domain in R2 and let f be a
holomorphic function in Ω. Let F be the set of critical values of the mapping
Log : f −1 (0) −→ R2 , and take a small open set V in Af r F . Let k be the
cardinality of f −1 (0) ∩ Log−1 (x) for x ∈ V . Then there exist smooth functions
φj , ψj , j = 1, . . . , k defined in V such that x 7→ (exp(x1 + iφj (x)), exp(x2 +
iψj (x))) are local inverses of Log, i.e. f −1 (0) ∩ Log−1 (V ) = ∪kj=1 {(exp(x1 +
iφj (x)), exp(x2 + iψj (x))); x = (x1 , x2 ) ∈ V }.
Theorem 18 ([25]). With notations as in the preceding paragraph, the Hessian
of Nf is given in V by the formula
k
1 X
∂ψj /∂x1
Hess Nf =
±
−∂φ
2π j=1
j /∂x1
∂ψj /∂x2
.
−∂φj /∂x2
(27)
The summands in the right hand side are positive definite matrices with determinant equal to 1.
Proof. Differentiating the integral (2) defining Nf with respect to x1 we obtain
Z
1
∂f /∂z1 dz1 dz2
∂Nf
= Re
∂x1
(2πi)2 Log−1 (x)
f (z)z2
Z
1
dz2
=
n(f (·, z2 ), x1 )
2πi log |z2 |=x2
z2
Z 2π
1
=
n(f (·, ex2 +iy2 ), x1 )dy2 .
2π 0
If f is a Laurent polynomial, n(f (·, z2 ), x1 ) is the number of zeros minus the
number of poles of the function z1 7→ f (z1 , z2 ) inside the disc {log |z1 | < x1 }. In
general, n(f (·, z2 ), x1 ) is an integer valued function such that n(f (·, z2 ), x1 ) −
n(f (·, z2 ), x01 ) is equal to the number of zeros of z1 7→ f (z1 , z2 ) in the annulus
{x01 < log |z1 | < x1 } when x01 < x1 . Hence, the integrand in the last integral is a
piecewise constant function with jumps of magnitude 1 at y2 = ψj (x). It follows
that the gradient of ∂Nf /∂x1 is given by a sum of terms ±(2π)−1 grad ψj . This
proves the first row of the identity (27), up to sign changes. The correct sign of
each term can be found by observing that n(f (·, ex2 +iy2 ), x1 ) is increasing as a
function of x1 , hence all the terms contributing to ∂ 2 Nf /∂x21 should be positive.
47
I
Polynomial amoebas
A similar computation involving ∂Nf /∂x2 proves the second row. However, we
have not yet shown that the choices of signs in the two rows are consistent.
We shall now prove that all the terms on the right hand side of (27) are
symmetric, positive definite matrices with determinant equal to 1. Take a point
x and an index j. Differentiating the expression f (ex1 +iφj (x) , ex2 +iψj (x) ) = 0
with respect to x1 and x2 yields the equations
∂φj
∂f ∂ψj
∂f
1+i
+ z2
z1
i
=0
∂z1
∂x1
∂z2 ∂x1
∂f ∂φj
∂ψj
∂f
z1
1+i
= 0.
i
+ z2
∂z1 ∂x2
∂z2
∂x2
Writing a = z1 ∂f /∂z1, b = z2 ∂f /∂z2, these equations have the solution
1
∂ψj /∂x1
∂ψj /∂x2
|a|2
Re(āb)
=
.
−∂φj /∂x1 −∂φj /∂x2
|b|2
Im(āb) Re(āb)
This matrix clearly has determinant 1. Changing the sign if Im(āb) < 0 we also
have that the diagonal elements are positive, so the matrix is positive definite.
Since we have already observed that the diagonal elements in the right hand
side of (27) must be positive, it follows that these matrices are positive definite
with determinant equal to 1.
Corollary 9. If f is a Laurent polynomial in two variables, then µf ≥ π −2 λ|Af
where λ denotes Lebesgue measure.
Proof. Since the set F of critical values of Log : f −1 (0) → R2 is a null set
for Lebesgue measure it suffices to prove the inequality in the complement of
this set. If A1 , A2 are 2 × 2 positive definite matrices, then it follows from
the inequality (20) √
that det(A1 + A2 ) = det A1 + det A2 + 2 det(A1 , A2 ) ≥
det√A1 det A2 . Repeated
use of this inequality leads to
det
A
+
det
A
+
2
2
p 1
√
det(A1 + . . . + Ak ) ≥ det A1 + . . . + det Ak . Applying this inequality to
the sum (27), which contains at least two terms for every x ∈ Af r F , yields
the result.
Theorem 19 ([25], [24]). If f is a Laurent polynomial in two variables, then
Area(Af ) ≤ π 2 Area(Pf ). When Pf has positive area, equality holds precisely if
Log−1 (x) intersects f −1 (0) in at most two points for every x ∈ R2 and there
exist constants a, b1 , b2 ∈ C∗ such that af (b1 z1 , b2 z2 ) is a polynomial with real
coefficients.
Proof. The inequality Area(Af ) ≤ π 2 Area(Pf ) follows immediately from
Corollary 9 and Theorem 15. Notice that equality holds if and only if µf =
π −2 λ|Af .
48
I
Polynomial amoebas
Suppose that Log−1 (x) ∩ f −1 (0) has at most two points for all x ∈ R2
and that af (b1 z1 , b2 z2 ) has real coefficients. Without loss of generality we may
assume that a = b1 = b2 = 1. Then the sum (27) contains precisely two terms
for all x ∈ Af and since complex conjugation of the coordinates leaves f −1 (0)
unchanged, we have φ2 = −φ1 , ψ2 = −ψ1 . This means that the two terms are
actually equal, and it follows that det Hess(Nf ) = π −2 and hence µf = π −2 λ
in Af r F . Now F is a null set with respect to Lebesgue measure, so if we can
show that µf (F ) = 0 it will follow that µf = π −2 λ|Af , which is what we want
to prove.
Let F̃ be a real algebraic curve containing the set of critical values of the
mapping f −1 (0) → R2 : (z1 , z2 ) 7→ (|z1 |2 , |z2 |2 ). Consider the product space
C2∗ × T2 and let π1 : C2∗ × T2 → R2 and π2 : C2∗ × T2 → T2 be defined
by π1 (z, t) = (|z1 |2 , |z2 |2 ) and π2 (z, t) = t. Let C = π1−1 (F̃ ) ∩ {f (z1 , z2 ) =
f (t1 z1 , t2 z2 ) = 0}. Since Log−1 (x) ∩ f −1 (0) is a finite set for every x ∈ R2 ,
it follows that the projection π1 : C → F̃ has finite fibers, hence C is a real
curve. It follows that π2 (C) is a null set in T2 , which means that the system of
equations f (z1 , z2 ) = f (t1 z1 , t2 z2 ) = 0 has no solutions in Log−1 (F ) for almost
all t ∈ T2 . It follows from Theorem 16 that µf (F ) = 0 as required.
Suppose now conversely that µf = π −2 λ|Af . First we show that f is irreducible. If K, L are compact convex sets in R2 , then it follows from the monotonicity properties of mixed volumes that Area(K + L) ≥ Area(K) + Area(L)
with strict inequality unless either K or L is a point or K and L are two parallel
segments. If f = gh is a nontrivial factorization of f we therefore have
Area(Af ) ≤ Area(Ag ) + Area(Ah ) ≤ π 2 (Area(Pg ) + Area(Ph ))
< π 2 Area(Pf )
contradicting the assumption that Area(Af ) = π 2 Area(Pf ).
It follows from Theorem 18 that Log−1 (0) ∩ f −1 (0) has at most two points
for all x outside F and that the two terms in the sum (27) are equal. After
a change of coordinates (z1 , z2 ) 7→ (z1 /b1 , z2 /b2 ) we may then assume that
φ2 = −φ1 , ψ2 = −ψ1 in a neighborhood of some point x ∈ Af r F (such points
exist by the assumption that Area(Af ) = π 2 Area(Pf ) > 0). But this means
that f (z̄1 , z̄2 ) vanishes on an open subset of f −1 (0). Since f is irreducible it
follows that f (z1 , z2 ) and f (z̄1 , z̄2 ) are equal up to a constant multiple. Hence
af (z1 , z2 ) has real coefficients for a suitable constant a.
It remains to be shown that Log−1 (x0 ) ∩ f −1 (0) has at most two points for
all x0 ∈ F . To do this, we consider two cases. Consider a discrete point of
Log−1 (x0 ) ∩ f −1 (0) and a small neighborhood U of it in f −1 (0). Now, either
Log(U ) contains a neighborhood of x0 , or there is an open half plane H with
x0 on its boundary such that Log−1 (x) ∩ U has two points for all x ∈ H near
x0 . If there are three discrete points in Log−1 (x0 ) this implies that one can
find x outside F such that Log−1 (x) ∩ f −1 (0) contains more than two points, a
contradiction. If instead Log−1 (x0 ) ∩ f −1 (0) contains a real curve, then it must
49
I
Polynomial amoebas
be of the form z α = c for some α ∈ Z2 and c ∈ C, otherwise µf would have
a point mass at x0 by Theorem 16. But then f has a factor z α − c, which is
impossible since Area(Pf ) is positive and f is irreducible.
7
Amoebas and real algebraic geometry
A relation between amoebas and real algebraic geometry has recently been discovered by Mikhalkin, who used amoebas to obtain results about the topology of
real algebraic curves (see [23]). In this section we briefly outline the background
to these results.
The study of the topology of real algebraic curves can be traced back to the
paper [13] by Harnack from 1876. The topology of real curves is also the subject
of Hilbert’s sixteenth problem. A more recent development in the theory is the
patchworking technique of Viro.
Throughout this section, P will denote a lattice polygon in R2 , and
F1 , . . . , Fk denote the edges of P , numbered in cyclic order. Let g be the number
of lattice points in the interior of P , and let the number of lattice points on Fj
be dj + 1. The polygon P determines a toric variety XP which is a compactification of the complex torus C2∗ . The closure of R2∗ in XP is a real toric variety,
which we denote RXP . (See section 2 for the notations being used for toric
varieties.)
Let f be a Laurent polynomial with real coefficients whose Newton polytope
is P , and let Vf = f −1 (0) be the hypersurface defined by f in C2∗ . Also let V f
be the closure of Vf in XP and let RVf and RV f be the intersection of RXP
with Vf and V f respectively.
Theorem 20 (Harnack [13], Khovanskii [18]). The genus of V f is equal to
g and the number of intersection points between V f and V (Fj ) is equal to dj .
Moreover, RV f has at most g + 1 connected components.
Definition 11 (see [17], [23], [24]). A Laurent polynomial f is said to define
a Harnack curve if the following conditions hold.
(i) RV f consists of g + 1 connected components.
(ii) One of these components can be divided into k consecutive arcs γ 1 , . . . , γk
such that γj intersects V (Fj ) in dj points and γj does not intersect V (Fl )
if j 6= l.
(iii) None of the other components of RV f intersects V (Fj ), j = 1, . . . , k.
Let A denote the set of lattice points in P and let cα be a real number for
every α ∈ A. Set
s(x) = max (cα + hα, xi)
α∈A
50
I
Polynomial amoebas
and let Σ0 be the subdivision of P obtained by the recipe in section 4. Assume
that the numbers cα have been chosen so that the 2-dimensional polygons in Σ0
are all triangles and that all points in A appear as vertices in Σ0 .
Let f t be a family of polynomials in RA such that log |fαt | = tcα and fαt is
negative if α ∈ 2Z2 and positive otherwise.
Theorem 21 (Harnack [13], Itenberg, Viro [17], Mikhalkin [23]). For
sufficiently large t, f t defines a Harnack curve.
Theorem 22 (Mikhalkin [24]). A nonsingular real polynomial f defines a
Harnack curve if and only if Log−1 (x) intersects Vf in at most two points for
all x ∈ R2 . In this case, RVf = Vf ∩ Log−1 (∂Af ).
Combining this with Theorem 19 we obtain
Corollary 10 ([24]). A nonsingular real polynomial f defines a Harnack curve
if and only if Area(Af ) = π 2 Area(Pf ).
Since Harnack curves exist for every polytope P by Theorem 21 we also have
Corollary 11 ([24]). The inequality Area(Af ) ≤ π 2 Area(Pf ) in Theorem 19
is sharp.
8
Amoebas of varieties of codimension greater
than 1
In this section we discuss a possible generalization of the material in previous
sections. The ideas presented are rather tentative, and we do not prove any
results. The central problem will be how the definitions of the main objects
we have studied might be carried over to a more general situation where the
hypersurface f −1 (0) is replaced by an arbitrary algebraic variety.
Let V be an algebraic variety in LC∗ . We assume that V is of pure dimension
k and let r = n − k be its codimension. By the amoeba of V we shall mean the
set AV = Log(V ).
8.1
The amoeba complement
If f is a Laurent polynomial, and V = f −1 (0) is a hypersurface, the amoeba
complement AcV consists of a number of connected components. Each such
component is convex, and in particular contractible. Therefore, the topology of
AcV is determined completely by the number of connected components.
The concept of the order of a complement component plays an important
role in the study of amoebas. Notice that the order of a complement component
cannot be defined only in terms of the hypersurface V . Indeed, if f is multiplied
by an arbitrary Laurent monomial z ν , then V is not changed, but the order
51
I
Polynomial amoebas
of each complement component is translated by the vector ν. On the other
hand, the difference between the orders of two components does not change,
so this difference may be possible to define in terms of V alone. This can be
done as follows. Let Eα and Eβ be two components of AcV whose orders are
α and β (with respect to the defining polynomial f ). Take arbitrary points
zα ∈ Log−1 (Eα and zβ ∈ Log−1 (Eβ ) and let a ∈ L. Let Cα and Cβ be the
oriented curves parametrized by ζ a zα and ζ a zβ where ζ runs along the unit
circle T in the counterclockwise direction. If B is an oriented surface in LC∗
with boundary Cα − Cβ , then hα − β, ai is equal to the number of intersection
points (counted with signs) between B and V .
We now generalize this construction to the case where the codimension r of
V is greater than 1. In this case the connected components of AcV are no longer
convex (in general) so AcV may have nontrivial homology groups. We willVfocus
here on Hr−1 (AcV , Z), and define a homomorphism ord : Hr−1 (AcV , Z) → r L∗ ,
which seems to be a natural generalization
of the order of complement compoVr
L. Since Log−1 (AcV ) is homeomorphic
nents. Let c ∈ Hr−1 (AcV , Z) and a ∈
(in a canonical way) to Log−1 (0) × AcV , ρr (a) ⊗ c defines a homology class in
H2r−1 (Log−1 (AcV ), Z). Let C be a (2r − 1)-cycle representing this homology
class. Since c is null homologous in LR , there exists a 2r-chain B in LC∗ whose
boundary is C. The number of intersection points between B and V depends
only on a and c and not on the choices made in selecting C and B. Moreover, the
dependence V
on a and c is bilinear. Hence ord(c) may be defined as the unique
r ∗
L such that hord(c),Vai is equal to the number of intersection
element in
r
points between B and V for all a ∈
L.
8.2
The Ronkin function
If we try to define the function Nf entirely in terms of V = f −1 (0) we face
the same problem as with the orders of complement components; when f is
multiplied by a monomial the function Nf is changed but V remains the same.
In order to rescue at least some fragment of the Ronkin function we note that
with respect to the Monge-Ampère operator these changes are of no significance.
e 1 , . . . , un−1 , Nf ) in terms of V
Theorem 17 gives an explicit formula for M(u
where u1 , . . . , un−1 are arbitrary convex functions; if Uj (z) = uj (Log z), then
Z
e 1 , . . . , un−1 , Nf )(E) = 1
M(u
ddc U1 ∧ . . . ∧ ddc Un−1 .
n! Log−1 (E)∩V
This can easily be generalized to the case where k = dim V is arbitrary. If we
define
Z
e V (u1 , . . . , uk )(E) = 1
M
ddc U1 ∧ . . . ∧ ddc Uk ,
n! Log−1 (E)∩V
e V (u1 , . . . , uk ) is a positive measure with support on AV depending multhen M
tilinearly on u1 , . . . , uk . The interpretation of µf1 ,...,fn given in Theorem 16 is
also open to generalizations.
52
I
Polynomial amoebas
8.3
The spine
It seems reasonable that the spine of AV should be a k-dimensional polyhedral
complex approximating AV . However, it is clear that the spine can in general
not be contained in the amoeba. This can be seen by considering the example
of a line in C3 .
We do not know if it is possible to define a spine with reasonable properties for arbitrary amoebas of higher codimension. Here we shall only mention
a formula, which gives some information about the spine in the hypersurface
case and which generalizes to varieties of higher codimension. Hence, let f be
a Laurent polynomial with amoeba Af and spine Sf . Moreover, let C be a line
segment in Rn , which is known to intersect the spine in exactly one point, and
whose endpoints are in the complement of the amoeba. Let α1 , . . . , αn−1 be linearly independent vectors normal to C, and note that dz αj /z αj is a well defined
differential form on Cn∗ , even if αj is not an integer vector. The differential form
ω=
dz α1
dz αn−1
∧
·
·
·
∧
z α1
z αn−1
is uniquely determined, up to a constant multiple, by the slope of C. Now we
claim that the jth component of the point p = Sf ∩ C is given by
R
log |zj |ω
.
(28)
pj = C R
ω
C
This formula has an obvious generalization to the higher codimension case.
Suppose that a spine SV can be defined for the amoeba of a k-dimensional
variety V and let r = n − k. Let C be a subset of an r-dimensional plane, that
intersects SV in a single point, and such that the boundary of C is contained in
AcV . Let α1 , . . . , αk be linearly indepent vectors normal to C, and let
ω=
dz α1
dz αk
∧ ···∧ α .
α
1
z
z k
A reasonable requirement is then that the jth component of p = SV ∩ C is given
by (28).
8.4
The Newton polytope
Finally, we propose an approach to generalized Newton polytopes. We suggest
that the Newton polytope of a variety of arbitrary dimension should be found in
the polytope algebra discovered by McMullen. Since many readers are probably
not familiar with this intriguing structure, we give a brief description here,
referring to [21] and [22] for details. A relation between the polytope algebra
and toric varieties which seems to be of interest in this context has been found
by Fulton and Sturmfels [10].
53
I
Polynomial amoebas
The polytope algebra Π is an abelian group generated by all polytopes in a
real vector space, which in our case will be LR . If P is a polytope, its class in Π
is denoted [P ]. The generators are subject to the relations [P + x] = [P ] for any
translation vector x (translation invariance) and [P ] + [Q] = [P ∪ Q] + [P ∩ Q]
whenever P, Q and P ∪ Q are polytopes (the valuation property). The polytope
algebra is the universal group for functions which are translation invariant and
satisfy the valuation property: If φ is a function from the set of all polytopes
into an abelian group which satisfies φ(P + x) = φ(P ) and φ(P ) + φ(Q) =
φ(P ∪ Q) + φ(P ∩ Q), then φ can be extended in a unique way to a group
homomorphism on Π.
A multiplication is defined on Π by the rule [P ] · [Q] = [P + Q], where P + Q
denotes the Minkowski (or vector) sum of P and Q. This operation makes Π
into a commutative ring.
Ln
There is a direct sum decomposition Π =
r=0 Ξr where each Ξr is an
abelian group. With respect to the multiplication in Π, Ξr · Ξs = Ξr+s (here
Ξr = 0 if r > n). Moreover, each Ξr with r ≥ 1 is in a natural way a real vector
space. Loosely speaking, Ξr carries information about the r-dimensional faces
of a polytope for r = 1, . . . , n.
The component Ξ0 is isomorphic to Z and is generated by the class of a
single point, while Ξn ∼
= R. The Ξn -component of a polytope is proportional
to its volume.
L
The subgroup Z1 = nr=1 Ξr is a nilpotent ideal in Π. For any p ∈ Z1 one
may therefore define
X
pj
log(1 + p) =
(−1)j−1
j
j≥1
and
exp p =
X pj
j≥0
j!
.
The functions log and exp are inverses of each other and satisfy exp(p1 + p2 ) =
exp p1 · exp p2 and log((1 + p1 )(1 + p2 )) = log(1 + p1 ) + log(1 + p2 ). For any
polytope P , [P ] − 1 belongs to Z1 and log[P ] is the Ξ1 -component of [P ]. The
set {log[P ]; P a polytope}, is a convex cone in Ξ1 .
If P is any polytope, we write h(P, x) = supξ∈P hξ, xi and Px = {ξ ∈
P ; hξ, xi = h(P, x)}.
If P is a polytope, then Π(P ) denotes the subalgebra of Π generated by all
polytopes Q which are Minkowski summands of P , that is P = tQ + R for some
t > 0 and some polytope R. Also, Ξr (P ) = Ξr ∩ Π(P ). If Q is a Minkowski
summand of P , there is a mapping F 7→ QF from the faces of P to the faces of
Q; if F = Px then QF = Qx . For any face F , QF is a Minkowski summand of
F and the mapping Q 7→ [QF ] ∈ Π(F ) is translation invariant and satisfies the
valuation property, hence it induces a homomorphism Π(P ) → Π(F ), which we
will denote p 7→ pF .
54
I
Polynomial amoebas
Suppose now that P is a lattice polytope. For any face F of P , let L∗F denote
the sublattice of L∗ which lies in the linear subspace of L∗R parallel to F , and let
LF be the dual lattice of L∗F , which is isomorphic to a quotient of L. If G is a
face of P and F is a facet of G, the set {a ∈ LG ; hξ−η, ai ≤ 0, ∀ξ ∈ G, η ∈ F } is a
semigroup isomorphic to Z≥0 . Let xF G denote the generator of this semigroup;
it is a kind of outer normal to the facet F of G. A real valued function w on
the set of all r-dimensional faces of P is called an r-weight. An r-weight w is
called a Minkowski weight if it satisfies the Minkowski relation
X
w(F )xF G = 0
F ⊂G
for every (r + 1)-dimensional face G of P , where the sum is taken over all facets
F of G. (We have deviated slightly here from McMullens treatment, since he
uses an inner product on the vector space to define the outer unit normal to a
facet of a polytope, while we use a lattice for this purpose.)
Now let Q be a Minkowski summand of P . For every r-dimensional face F
of P , the lattice L∗F determines a volume form on subspaces parallel to F . Let
w(F ) be the volume of QF with respect this volume form. It can then be shown
that w is a Minkowski weight. Similarly, any p ∈ Π(P ) determines a Minkowski
r-weight on P for any r = 0, . . . , n. Moreover, if P is a simple polytope (that
is, exactly n facets meet at every vertex of P ), then McMullen has shown that
the set of r-weights on P is isomorphic, via the above construction, to Ξr (P ).
We are now ready to define the Newton polytope of a variety V . Let P be a
simple polytope, and assume that the toric variety XP is nonsingular. Let V be
a k-dimensional subvariety of XP which intersects V (F ) transversely for every
face F of P . Let r = n − k be the codimension of V and define an r-weight on
P by letting w(F ) be the number of intersection points between V and V (F ).
Then w is a Minkowski weight, which can be seen as follows. If G is an (r + 1)dimensional face ofP
P and α ∈ L∗G , then z α defines
P a rational function on V (G)
whose divisor is − F ⊂G hα, xF G iV (F ). Hence F ⊂G hα, xF G iw(F ) is equal to
the number of poles minus the number of zeros of z α on V ∩ V (G), which must
be 0. Since this is true for all α ∈ L∗G , it follows that w satisfies the Minkowski
relations. Hence w can be identified with an element in Ξr (P ), which can be
thought of as a generalized Newton polytope. If f is a Laurent polynomial
whose Newton polytope is a Minkowski summand of P , the construction may
be applied to the closure of f −1 (0) in XP . The generalized Newton polytope
obtained in this case is log[Pf ], from which Pf can be reconstructed up to
translations.
9
Examples of amoebas
To calculate explicitly the amoeba and Ronkin function of a given polynomial is
very messy in all but the simplest cases. To motivate and exemplify the theory
55
I
Polynomial amoebas
in the previous sections we give here some examples of polynomials for which
certain calculations can be carried out without too much effort.
Example 1. First we consider a polynomial in one variable f (z) = f0 + f1 z +
. . .+fm−1 z m−1 +z m = (z +a0 ) . . . (z +am ) where it is assumed that |a1 | ≤ . . . ≤
|am |. The amoeba of f is then the discrete point set {log |a1 |, . . . , log |am |}. A
typical complement component of Af is an interval (log |aα |, log |aα+1 |) for some
α = 1, . . . , m − 1. The order of this component is α. In addition, there are the
two unbounded components (−∞, log |a1 |) and (log |am |, +∞), whose orders are
0 and m respectively.
Suppose Acf has a component of order α and let x be a point in that component. Then it follows that
Z
f (z)
Φα (f ) =
log α dη(z)
−1
z
Log (x)
Z
Z
α
m
X
X
z + aj dz
dz
=
+
log
log(z + aj )
z
2πiz
2πiz
j=1 log |z|=x
j=α+1 log |z|=x
=
m
X
log aj
j=α+1
= log(aα+1 . . . am ).
Note that Φα has a branched analytic continuation to all polynomials without
multiple roots. The branches of this continuation correspond to various (m−α)element subsets of {1, . . . , m}. It is amusing to note that the sum of all branches
of exp Φα (f ) is equal to fα .
Recall that for any Laurent polynomial f we have
Nf (x) ≥ max(Re Φα (f ) + hα, xi)
α
with equality in the closure of Acf . If f is a polynomial in one variable, then Acf
is dense in R and so we have
Nf (x) = max (log |aα+1 . . . am | + hα, xi).
α
This is just a different formulation of the classical Jensen formula
1
2π
Z
2π
0
iθ
log |f (re )| dθ = log |f (0)| +
where α is the largest index such that |aα | < r.
56
α
X
j=1
log
r
|aα |
I
Polynomial amoebas
Example 2. Let us write up defining equations and inequalities for the set
UαA in the simplest nontrivial case, namely for quadratic polynomials in one
variable. Let f (z) = f0 + f1 z + f2 z 2 . The zeros of f are
p
−f1 ± f12 − 4f0 f2
2f2
p
and these have the same modulus precisely if f1 and i f12 − 4f0 f2 are linearly
dependent over R. This happens precisely if f12 and 4f0 f2 − f12 are positive
multiples of each other, or equivalently
f¯12 (4f0 f2 − f12 ) ≥ 0.
{0,1,2}
Hence f ∈ U1
precisely if this relation is not satisfied.
Example 3. If A ⊂ L∗ is affinely independent, and f ∈ CA , then Af =
{x; mα (f ; x) ≥ 1, ∀α ∈ A}. When A = {0, e1 , . . . , en } where e1 , . . . , en is the
standard basis for Zn , Af is called a hyperplane amoeba. When f is a product
of such linear factors, Af is called an arrangement of hyperplane amoebas.
Arrangements of hyperplane amoebas were studied extensively in [9].
Example 4. Next we consider polynomials in two variables of the form f (z) =
a+z1 +z2 +z1 z2 , assuming to begin with that a is an arbitrary complex constant.
It can then be shown that the amoeba of f is the set of points satisfying
|a|4 − 2|a|2 e2x1 + e4x1 − 2|a|2 e2x2 − (2 − 8 Re a + 2|a|2 )e2x1 +2x2
− 2e4x1 +2x2 + e4x2 − 2e2x1 +4x2 + e4x1 +4x2 ≤ 0. (29)
We now specialize to the case where a is real. It turns out that the amoeba
looks rather different depending on the sign of a. Consider first the case a < 0.
The inequality (29) can then be written
(ex1 +x2 − ex1 − ex2 − |a|)(ex1 +x2 − ex1 + ex2 + |a|)
× (ex1 +x2 + ex1 − ex2 + |a|)(ex1 +x2 + ex1 + ex2 − |a|) ≤ 0.
Each of the factors vanishes on the boundary of one of the complement components of the amoeba. Moreover, Log−1 (x) intersects f −1 (0) in at most two
points for every x so µf = π −2 λ|Af and the area of Af is π 2 by Theorem 19.
If a > 0, there is a similar factorization
(ex1 +x2 − ex1 − ex2 + |a|)(ex1 +x2 − ex1 + ex2 − |a|)
× (ex1 +x2 + ex1 − ex2 − |a|)(ex1 +x2 + ex1 + ex2 + |a|) ≤ 0.
57
I
Polynomial amoebas
Figure 2: Amoebas of the polynomial f (z) = a + z1 + z2 + z1 z2 for a =
−5, −1, −1/5, 1/5, 1 and 5 together with their spines and dual subdivisions of
the Newton polytope.
Notice that the fourth factor is always positive, while the first factor vanishes
on the boundaries of two complements components (those of orders (0, 0) and
(1, 1) if a < 1 and those of orders (0, 1) and (1, 0) if a > 1). The remaining
factors define two curves, each of which constitutes part of the boundary of
two different complement components. The curves intersect at their common
point of inflection (log a/2, log a/2). Moreover, Log−1 (x) intersects f −1 (0) in
at most two points except when x = (log a/2, log a/2). For this special x,
Log−1 (x) ∩ f −1 (0) is a real curve. It follows from Theorem 16 and Theorem 18
that the measure µf is equal to π −2 λ|Af plus a point mass at (log a/2, log a/2).
The size of the point mass can be computed explicitly by means of Theorem 16,
and one finds that it is
Z
16 π/2
µa = 2
arcsin(a±1/2 cos t) dt.
π 0
where the positive sign is chosen in the exponent if a < 1 and the negative sign
otherwise. Consequently, the area of the amoeba is π 2 (1 − µa ). The area of teh
amoeba can also be computed directly.
If a = 1, then f (z) = (z1 + 1)(z2 + 1) and the amoeba is the union of two
lines.
Example 5. Consider Laurent polynomials in one variable of the form f (z) =
g(z + z −1) − a, where g is an arbitrary polynomial and a is a constant. We shall
determine the set of a for which Acf has a component of order 0. Note that
f (z −1 ) = f (z), hence the amoeba of f is symmetric with respect to reflection in
the origin. In particular, if Acf has a component of order 0, then it is mapped
onto itself by this reflection. Hence a complement component of order 0, if
it exists, must contain the origin. Conversely, if a complement component of
order α contains the origin, then reflection in the origin maps it to a complement
component of order −α. Since these components have nonempty intersection,
58
I
Polynomial amoebas
it follows that α = 0. We conclude that Acf has a component of order 0 if and
only if 0 ∈ Acf . Now, 0 ∈ Af means that f has a zero on the unit circle. Since
z 7→ z + z −1 maps the unit circle onto the interval [−2, 2], it follows that Acf
has a component of order 0 precisely if a ∈
/ g([−2, 2]). Keeping g fixed and
letting a vary, we see that the set {a; E0 (f ) 6= ∅} may have any (finite) number
of connected components. The complement of this set is of course connected,
in accordance with Theorem 14.
Example 6. Another class of polynomials on which certain computations can
easily be carried out explicitly is polynomials of the form f (z) = 1 + z1n+1 +
. . . + znn+1 + az1 . . . zn where a is an arbitrary complex constant. The Newton
polytope of f is a simplex with precisely one lattice point, namely (1, . . . , 1),
in its interior. It follows from Theorem 12 or Theorem 11 that the only lattice
points in Pf which can occur as orders of complement components, are the
vertices and the interior point (1, . . . , 1). Now, if α is a vertex of Pf there is
always a complement component of order α. We shall compute the set of a for
which Acf has a component of order (1, . . . , 1).
Figure 3: Amoeba and triangulated Newton polytope of the polynomial f (z) =
1 + z13 + z23 + az1 z2 for a = −6 and the set of a for which E(1,1) (f ) is empty.
For reasons of symmetry, as in the previous example, it can be shown that
a component of order (1, . . . , 1) must necessarily contain the origin, and conversely, if a complement component contains the origin, then its order must
be (1, . . . , 1). It is also clear that 0 ∈ Af precisely if −a belongs to the set
Kn = {t0 + . . . + tn ; |t0 | = . . . = |tn | = t0 . . . tn = 1} ⊂ C. This set is contained
in the closed disc of radius n + 1 and contains the disc of radius n − 1 centered
at the origin. The boundary of Kn has n + 1 cusps; the corresponding values
of a give rise to polynomials defining singular hypersurfaces, and are branching points for Φ(1,...,1) (f ). Finally, we note that the power series expansion of
Φ(1,...,1) (f ) computed in Theorem 6 takes the simple form
Φ(1,...,1) (f ) = log a −
X ((n + 1)k − 1)!
(−a)(n+1)k .
(k!)n+1
k≥1
59
I
Polynomial amoebas
P
Example 7. If f (z) = α∈A fα z α is a Laurent polynomial where A has no
more than 2n elements, and these are in sufficiently general position, then by the
proof of Theorem 12, the order of a complement component of Af is determined
by the dominating term in f . The assumption about general position implies in
particular, that no three points in A are collinear. The following example shows
that the situation becomes quite different if A is allowed to have three collinear
points.
P
Let f be a Laurent polynomial of the form f (z) = a0 + α∈A aα (z α − z −α ).
Here we assume that A is a finite set not containing the origin, and that all
the coefficients aα except a0 are real. For reasons of symmetry, a complement
component of Af containing the origin must have order 0, and conversely, a
component of order 0 must contain the origin. Now, if z ∈ Log−1 (0), then
z α + z −α is real for any α. It follows that Acf has a complement component of
order 0 as soon as a0 is not real, no matter how small it is.
10
Some open problems
Here are a few seemingly interesting and open problems related to the subject
of this thesis. After each problem we give some comments including elementary
observations, known results of a similar nature and some guesses about solutions.
Problem 1. Let A ⊂ Zn be a finite set and let α ∈ Zn be a point. Find
a necessary and sufficient condition for the existence of a Laurent polynomial
f ∈ CA with Eα (f ) 6= ∅.
Theorem 11 gives one necessary condition and one sufficient condition. However, the gap between the two conditions is usually very large. For certain simple
sets A the complete answer is given by Theorem 12. A rather wild guess would
be that the second condition in Theorem 11 is also a necessary condition.
Problem 2. Given an integer polytope P ⊂ Z2 , what is the minimal area of
the amoeba Af given that Pf = P ?
This problem was suggested by Oleg Viro and communicated to the author
by Grigory Mikhalkin. If P is a zonotope, that is the Minkowski sum of line
segments, then f can be taken to be a product of binomials and the area of
the amoeba is zero in this case. If P is a triangle whose area is 1/2, the area
of Af will always be π 2 /2, whereas if P is an arbitrary triangle and the only
nonzero coefficients in f are those corresponding to the vertices of P , then
Area(Af ) = π 2 /(4 Area(P )). In view of this it seems reasonable to conjecture
that Area(Af ) ≥ c/ Area(Pf ) for some constant c > 0 unless Pf is a zonotope.
The constant c can be no greater than π 2 /4. However, this estimate is probably
not sharp for most polytopes P .
60
I
Polynomial amoebas
Problem 3. Let f0 , . . . , fm be Laurent polynomials. Classify all convergent
fractional Laurent series g satisfying the equation f0 + f1 g + . . . + fm g m = 0.
A fractional Laurent series is an infinite linear combination of fractional
Laurent monomials z α , where α ∈ L∗1 and L1 ⊂ L is a sublattice of the same
rank as L.
If m = 1 and f0 = −1, Theorem 4 associates every such g with a connected
component of Acf1 , and each such component is associated with a lattice point in
Pf1 . Hence there is a bijective correspondence between the convergent Laurent
series g and a subset of L∗ ∩ Pf1 .
As demonstrated in the introduction, the Laurent series associated with a
vertex of Pf1 are most easily computed. The analogous series g for the equation
f0 + f1 g + . . . + fm g m = 0 were classified by McDonald in [20]. There it was
shown that they correspond in a natural way to certain edges of Pf ⊂ L∗R × R,
where f (z, t) = f0 (z) + f1 (z)t + . . . + fm (z)tm . It would be nice to find a similar
generalization of the non-vertex lattice points.
61
I
11
Polynomial amoebas
List of notations
Symbol
Explanation
Defined on page
aff A
Af
Acf
Area
C∨
C∗
CA
cone(F, P )
conv
dc
Eα
η
Hess
int
λ
Log
M
e
M
mα (f )
mα (f ; x)
µf
µf1 ,...,fn
Nf
nc(F, P )
Pf
Φα
ρk , ρk
relint
Sf
T
Tn
UαA
vert P
Vol
XP , X Σ
Affine lattice generated by A
Amoeba of f
Complement of amoeba
Area in R2
Dual of cone C
C r {0}
P
The space of Laurent polynomials α∈A fα z α
Cone of vectors from F into P
Convex hull
¯
(∂ − ∂)/2πi
Complement component of order α
Haar measure
Hessian matrix
Interior of a set
Lebesgue measure
Monge-Ampère operator
Mixed Monge-Ampère operator
inf
P x mα (f ; x)
β6=α |fβ /fα | exphβ − α, xi
M Nf
e f 1 , . . . , Nf n )
M(N
Ronkin function of f
Normal cone to P at F
Newton polytope of f
Relative interior of a convex set
Spine of Af
Unit circle in C
T × . . . × T (n factors)
{f ∈ CA ; Eα (f ) 6= ∅}
Set of vertices of P
Volume in Rn
Toric varieties
62
18
18
21, 32
20
21, 32
42
23
17
40
17
40
41
24
24
44
44
19
21
19
26
19
33
35
21
I
Polynomial amoebas
References
[1] A. D. Aleksandrov: Zur Theorie von konvexen Körpern IV: Die gemischten
Diskriminanten und die gemischten Volumina, Matem. Sb. SSSR 3 (1938),
227–251.
[2] E. Bedford, A. Taylor: The Dirichlet problem for a complex Monge-Ampère
equation, Invent. Math. 37 (1976), 1–44.
[3] D. Bernstein: The number of roots of a system of equations, Functional
Anal. Appl. 9 (1975), 183–185.
[4] V. I. Danilov: The geometry of toric varieties, Russian Math. Surveys, 33:2
(1978), 97–154.
[5] J. Duistermaat, W. van der Kallen: Constant terms in powers of a Laurent
polynomial, Indag. Math. 9 (1998), 221–231.
[6] G. Ewald: Combinatorial convexity and algebraic geometry, SpringerVerlag, New York, 1996.
[7] G. Faber: Über die zusammengehörigen Konvergenzradien von Potenzreihen mehrerer Veränderlicher, Math. Ann. 61 (1905), 289–324.
[8] M. Forsberg: Amoebas and Laurent series, Doctoral thesis, KTH Stockholm, 1998.
[9] M. Forsberg, M. Passare, A. Tsikh: Laurent determinants and arrangements of hyperplane amoebas, Adv. in Math. 151 (2000), 45–70.
[10] W. Fulton, B. Sturmfels: Intersection theory on toric varieties, Topology
36 no. 2 (1997), 335–353.
[11] I. Gelfand, M. Kapranov, A. Zelevinsky: Generalized Euler integrals and
A-hypergeometric functions, Adv. in Math. 84 (1990), 255–271.
[12] I. Gelfand, M. Kapranov, A. Zelevinsky: Discriminants, resultants and
multidimensional determinants, Birkhuser, Boston, 1994.
[13] A. Harnack: Über Vieltheiligkeit der ebenen algebraischen Curven, Math.
Ann. 10 (1876), 189–199.
[14] F. Hartogs: Zur Theorie der analytischen Funktionen mehrerer unabhängiger Veränderlichen, insbesondere über die Darstellung derselben
durch Reihen, welche nach Potenzen einer Verändrlichen fortschreiten,
Math. Ann. 62 (1906), 1–88.
[15] L. Hörmander: Notions of convexity, Birkhäuser, Boston, 1994.
63
I
Polynomial amoebas
[16] A. Henriques: An analogue of convexity for complements of amoebas of
varieties of higher codimensions, Preprint, Berkeley, May 2001.
[17] I. Itenberg, O. Viro: Patchworking algebraic curves disproves the Ragsdale
conjecture, Math. Intelligencer 18 (1996), no. 4, 19–28.
[18] A. G. Khovanskii: Newton polyhedra and the genus of complete intersections, Functional Anal. Appl. 12 (1978), 38–46.
[19] C. Kiselman: The partial Legendre transformation for plurisubharmonic
functions, Invent. Math. 49 (1978) 137–148.
[20] J. McDonald: Fiber polytopes and fractional power series, J. Pure Appl. Algebra, 104 (1995), 213–233.
[21] P. McMullen: The polytope algebra, Adv. in Math., 78 (1989), 76–130.
[22] P. McMullen: Separation in the polytope algebra, Beitrge Algebra Geometrie, 34 (1993), 15–30.
[23] G. Mikhalkin: Real algebraic curves, moment map and amoebas, Ann. of
Math., 151 (2000),no. 1, 309–326.
[24] G. Mikhalkin, H. Rullgård: Amoebas of maximal area, International Mathematics Research Notices, 9 (2001), 441–451.
[25] M. Passare,
H. Rullgård:
Amoebas,
Monge-Ampère measures, and triangulations of the Newton polytope. Preprint:
http://www.math.su.se/reports/2000/10. To appear in Duke Math.
J.
[26] A. Rashkovskii: Indicators for plurisubharmonic functions of logarithmic
growth, Indiana Univ. Math. J. 50 (2001), no. 3, 1433–1446.
[27] J. Rauch, A. Taylor: The Dirichlet problem for the multidimensional
Monge-Ampre equation, Rocky Mountain J. Math. 7 (1977), 345–364.
[28] L. Ronkin: Introduction to the theory of entire functions of several variables, Translations of mathematical monographs, AMS, Providence, 1974.
[29] L. Ronkin: On zeros of almost periodic functions generated by holomorphic functions in a multicircular domain, in Complex Analysis in Modern
Mathematics, Fazis, Moscow, 2000, pp. 243–256.
[30] H. Rullgård: Stratification des espaces de polynômes de Laurent et structure de leurs amibes, C. R. Acad. Sci. 2000, 355–358.
[31] T. Sadykov: Hypergeometric systems of differential equations and amoebas
of rational functions, Preprint: Potsdam 1999.
64
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Polynomial amoebas
[32] O. Viro: Real algebraic curves: Constructions with controlled topology,
Leningrad Math. J., 1 (1990), 1059–1134.
65
II
Differential equations in the
complex plane
Differential equations in the complex plane
1
II
Some general remarks
Let Ω ⊂ C be a domain in the complex plane, and consider a linear ordinary
differential equation of order k > 1
k
X
sj (z; λ)f (j) (z) = 0
(1)
j=0
in the unknown function f (z). Here sj (z; λ) are holomorphic functions of z ∈ Ω
depending on a positive real parameter λ. Moreover, we assume that there
exist holomorphic functions Sj (z) defined in Ω, not all identically 0, such that
λj sj (z, λ) → Sj (z) when λ → +∞. We want to address the following question.
What can we say about the solutions to (1) in the limit when λ → +∞?
We also consider the following problem. Find all complex valued functions
C(z) (not necessarily holomorphic) defined in Ω, such that

k

X



Sj (z)C(z)j = 0
(2)
j=0


∂C


≥ 0.
∂ z̄
The first equation says that, at a generic point z ∈ Ω, the function C may
take any of k different values. As long as we can consistently choose one of
these branches at different points, the function C will be holomorphic and the
inequality is trivially satisfied. The restriction imposed by the inequality is that,
when C(z) jumps between different branches, this must be done in such a way
that the distributional derivative ∂C/∂ z̄ is a positive measure.
Now we conjecture that the following holds, under suitable restrictions on the
functions sj . If f (z) is a solution to (1) for some large λ, then there exists a
solution C(z) to (2) such that
f 0 (z)
≈ C(z),
λf (z)
(3)
and the larger value of λ we take, the better the approximation can be made.
To make this statement into a theorem, it is of course necessary to define the
precise meaning of the approximate equality.
Closely related to (3), are two other approximations. For a holomorphic
function f and a parameter λ satisfying (1), let µf denote the measure on
Ω obtained by placing a point mass of size 1/λ at each zero of f , counting
69
II
Differential equations in the complex plane
zeros with multiplicity. Then differentiating both sides of (3) with respect to z̄,
suggests that
1 ∂C
µf ≈
.
(4)
π ∂ z̄
On the other hand, integrating both sides of (3), suggests that
1
log |f (z)| ≈ u
λ
where u is a subharmonic function satisfying
j
k
X
∂u
Sj (z) 2
=0
∂z
j=0
(5)
(6)
for almost all z. One interesing aspect of this condition is that if u1 and u2 are
subharmonic functions satisfying (6), then u1 + c and max(u1 , u2 ) are also such
functions. Therefore, it is possible that all subharmonic solutions of (6) can be
generated from a small number of solutions, using maximum and addition of
constants.
In the next section we carry out these considerations in detail and give proofs
for a special class of differential equations. In these cases, Ω is the whole complex
plane and both the coefficients sj and the solutions f are polynomials in z. The
treatment is simplified by the fact that the space of solutions of (1) is at most
one dimensional for each value of λ.
2
On polynomial eigenfunctions for a class of
differential operators 1
2.1
Introduction
Jacobi polynomials are solutions of the differential equation
(z 2 − 1)f 00 (z) + (az + b)f 0 (z) + cf (z) = 0,
(7)
where a, b, c are constants satisfying a > b, a + b > 0 and c = n(1 − a − n) for
some nonnegative integer n. It is a classical fact that the zeros of the Jacobi
polynomials lie in p
the interval [−1, 1], and that their density in this interval is
proportional to 1/ 1 − |z|2 in the limit when the degree n tends to infinity. The
usual proof of this statement involves the observation that, for fixed a and b, the
Jacobi polynomials constitute an orthogonal system of polynomials with respect
to a certain weight function on the interval [−1, 1]. The desired conclusion then
follows from the general theory of orthogonal systems of polynomials.
1 This section is joint work with Tanja Bergkvist and has been published in Math. Research
Letters 9 (2002), 153 – 171.
70
Differential equations in the complex plane
II
The following appears to be a natural generalization of the differential equation (7). Let k ≥ 2 be an integer, and let Q0 , . . . , Qk be polynomials in one
complex variable satisfying deg Qj ≤ j with equality when j = k. Moreover, we
make a normalization by assuming that Qk is monic. Consider the differential
operator
k
X
TQ (f ) =
Qj f (j)
(8)
j=0
(j)
where f
denotes the jth derivative of f . Operators of this type appear for
example in the theory of Bochner-Krall systems, see [1]. This operator was
studied by G. Masson and B. Shapiro in [4]. Particular attention was given
the more special operators T 0 (f ) = Qk f (k) and T 00 (f ) = (d/dz)k (f (z)Qk (z)).
These are indeed special cases of (8) obtained by taking Qj = 0 or Qj =
(k−j)
k
respectively, for j = 0, . . . , k − 1. The following result, which shows
j Qk
that TQ has plenty of polynomial eigenfunctions, was proved for the operators
T 0 and T 00 in [4].
Theorem 1. For all sufficiently large integers n there is a unique constant λ n
and a monic polynomial pn of degree n which satisfy
TQ (pn ) = λn pn .
(9)
Moreover, we have λn /n(n − 1) . . . (n − k + 1) → 1 when n → ∞.
G. Masson and B. Shapiro made a number of striking conjectures, based on
numerical evidence, about the zeros of the eigenpolynomials pn . They also
observed that when k > 2, the sequence pn is in general not an orthogonal
system of polynomials, so they cannot be studied by means of the extensive
theory known for such systems.
The goal of this note is to prove some of the conjectures in [4]. More precisely,
we shall show that in the limit when n → ∞, the zeros of pn are distributed
according to a certain probability measure. This probability measure depends
only on the “leading polynomial” Qk and may be independently characterized
in the following way.
Theorem 2. Let Qk be a monic polynomial of degree k. Then there exists a
unique probability measure µQk with compact support whose Cauchy transform
Z
dµQk (ζ)
C(z) =
z−ζ
satisfies C(z)k = 1/Qk (z) for almost all z ∈ C.
71
II
Differential equations in the complex plane
We record some properties of the measure µQk which will be encountered in
the proof of Theorem 2. Let supp µ denote the support of a measure µ. Also,
let
Z
Ψ(z) = Qk (z)−1/k dz
be a primitive function of Qk (z)−1/k . At this point, we think of Ψ as a locally
defined function in any simply connected domain where Qk does not vanish. The
choice of a branch of Qk (z)1/k and an integration constant is of no importance
here. As need arises, specifications will be made concerning these choices.
Theorem 3. Let Qk and µQk be as in Theorem 2. Then supp µQk is the union
of finitely many smooth curve segments, and each of these curves is mapped to
a straight line by the mapping Ψ. Moreover, supp µQk contains all the zeros of
Qk , is contained in the convex hull of the zeros of Qk and is connected and has
connected complement.
If p is a polynomial of degree n, we can construct a probability measure µ
by placing a point mass of size 1/n at each zero of p. We will call µ the root
measure of p. Our main result is
Theorem 4. Let pn be the monic eigenpolynomial of degree n of the operator
TQ and let µn be the root measure of pn . Then µn converges weakly to µQk when
n → ∞.
To illustrate, we show the zeros of the eigenpolynomial p40 for the degree
5 operator TQ with Q5 (z) = z(z − 1 + i)(z − 5)(z − 2 − 4i)(z − 4 − 4i) and
Q0 = · · · = Q4 = 0. Large dots represent the zeros of Q5 (which are, in this
case, also zeros of pn ) and small dots represent (the remaining) zeros of p40 . It
is remarkable how well the zeros of the eigenpolynomial line up along the curves
predicted by our results. Notice also how these curves are straightened out by
the mapping Ψ.
This paper is organized as follows. In section 2.2 we compute the matrix
for the operator TQ with respect to the basis of monomials 1, z, z 2, . . ., and use
this to prove Theorem 1. Section 2.3 contains a proof of the uniqueness part
of Theorem 2. Along the way, we also prove essentially all the statements in
Theorem 3. In section 2.4 we recall some basic facts on the weak topology
of measures in the complex plane and on logarithmic potentials and Cauchy
transforms. We also outline the connection of these concepts to root measures
of polynomials and prove a general result on the relation between the zeros of
a polynomial and those of its derivative. In the final section 2.5 we apply the
ideas from the previous section to give a proof of Theorem 4. The existence
part of Theorem 2 is also a consequence of this proof.
72
II
Differential equations in the complex plane
Figure 1: Zeros of the polynomial Q5 and the eigenpolynomial p40 (left) and
the image of these zeros under a branch of the mapping Ψ.
Acknowledgements. The authors are sincerely grateful to Harold Shapiro
for highly valuable comments and discussions about the work behind the present
paper. We would also like to thank Gisli Masson and Boris Shapiro for introducing us to the problem and for their support during our work.
2.2
Calculation of the matrix
Recall that the differential operator TQ is defined by
TQ = Q k
dk−1
d
dk
+ Qk−1 k−1 + · · · + Q1
+ Q0
k
dz
dz
dz
where the Qm are polynomials
Pn such that deg Qm ≤ m for m = 0, . . . , k and
deg Qk = k. P
Let pn (z) = i=0 an,i z i be a monic polynomial of degree n and
j
let Qm (z) = m
j=0 qm,j z . Using these notations we get
TQ (pn ) =
k
X
m=0
=
k
X
Qm ·
n X
m=0 s=0
=
n X
k
X
s=0
m=0
X
k X
m
n
X
i!
dm
i−m
j
p
=
z
q
z
·
a
·
n
m,j
n,i
dz m
(i − m)!
m=0 j=0
i≥m
X
s=j+i−m
m≤i≤n
0≤j≤m
X
s=j+i−m
m≤i≤n
0≤j≤m
qm,j · an,i ·
qm,j
i!
zs
(i − m)!
i!
zs
· an,i ·
(i − m)!
With pn monic and TQ (pn ) = λn ·pn = λn z n +λn ·an,n−1 z n−1 +. . .+λn ·an,0 ,
finding the eigenvalue λn amounts to finding the coefficient at z n in TQ (pn ).
dm
Note that deg Qm dz
m pn ≤ m + n − m = n with equality if and only if deg Q m =
73
II
Differential equations in the complex plane
m. Thus we can assume that pn = z n (since any lower degree terms of pn will
result in terms of degree lower than n in TQ (pn ). We therefore consider
k
X
k
X
dm n
n!
z
z n−m
=
Qm ·
m
dz
(n
−
m)!
m=0
m=0
k X
m
X
n!
n−m
j
z
=
qm,j z ·
(n − m)!
m=0
j=0
TQ (z n ) =
=
Qm ·
k X
m
X
m=0
qm,j
j=0
n!
j+n−m
z
.
·
(n − m)!
Setting j = m we get
λn z n =
k
X
m=0
qm,m ·
n!
zn
(n − m)!
=⇒
λn =
k
X
m=0
qm,m ·
n!
.
(n − m)!
Lemma 1. For n ≥ 1 the coefficient vector X of pn with components
an,0 , . . . , an,n−1 satisfies the linear system M X = Y , where Y is a vector and
M is an upper triangular matrix, both with entries expressible in the coefficients
qm,j (see below).
Proof. The relation
is equivalent to
n X
k
X
s=0
m=0
TQ (pn ) = λn · pn
X
s=j+i−m
m≤i≤n
0≤j≤m
qm,j · an,i ·
n
X
i!
z s = λn ·
an,s z s .
(i − m)!
s=0
With j = m + s − i the condition j ≤ m gives i ≥ s and the condition j ≥ 0
results in m ≥ i − s. Therefore the above system will be equivalent to
n X
X
X
i!
· an,i − λn · an,s z s = 0.
qm,m+s−i ·
(i − m)!
s=0
s≤i≤n
i−s≤m≤min(i,k)
For each s we have
X
s≤i≤n
or, equivalently,
X
s≤i≤n−1
X
i−s≤m≤min(i,k)
X
qm,m+s−i ·
i−s≤m≤min(i,k)
i!
· an,i − λn · an,s = 0
(i − m)!
qm,m+s−i ·
74
i!
· an,i − λn · an,s =
(i − m)!
II
Differential equations in the complex plane
X
=
n−s≤m≤min(n,k)
qm,m+s−n ·
n!
· an,n
(n − m)!
where an,n = 1. The left-hand side of the above equation corresponds to the
(s + 1)st row in M multiplied by the column vector X, and the right-hand side
represents the (s + 1)st row in Y . The n × n matrix M is thus constructed for
0 ≤ s ≤ n − 1 and 0 ≤ i ≤ n − 1. Thus the entries of M are given by
X
Ms+1,i+1 =
i−s≤m≤min(i,k)
qm,m+s−i ·
i!
− λn · δi,s
(i − m)!
(10)
where δ denotes the Kronecker delta. The condition i ≥ s implies that the
matrix M will be upper triangular.
We can now prove Theorem 1. For our operator TQ with qk,k = 1 we get
Pk
n!
λn
m=0 qm,m · (n−m)!
=
n(n − 1) . . . (n − k + 1)
n(n − 1) . . . (n − k + 1)
=
n!
n!
n!
n!
n!
+ q1,1 (n−1)!
+ q2,2 (n−2)!
+ . . . + qk−1,k−1 (n−k+1)!
+ qk,k (n−k)!
q0,0 n!
n(n − 1) . . . (n − k + 1)
q0,0
q1,1
qk−1,k−1
=
+
+ . . .+
+ qk,k .
n(n − 1) . . . (n − k + 1) (n − 1) . . . (n − k + 1)
(n − k + 1)
Thus
lim
n→∞
λn
= qk,k = 1.
n(n − 1) . . . (n − k + 1)
To prove uniqueness, we show that the determinant of the matrix M constructed
above is non-zero for sufficiently large values of n. Since the matrix is upper
triangular its determinant equals the product of the diagonal elements. Thus it
suffices to prove that for sufficiently large n every diagonal element is non-zero.
The diagonal element Mi+1,i+1 of M is obtained by letting i = s in (10) and so
equals
X
i!
qm,m ·
− λn
(i − m)!
0≤m≤min(i,k)
for i = 0, . . . , n − 1. But the last expression equals λi − λn . Indeed, if i ≥ k
then
X
X
i!
i!
=
= λi .
qm,m ·
qm,m ·
(i − m)!
(i − m)!
0≤m≤k
0≤m≤min(i,k)
If i < k then this is again true since by definition i!/(i − m)! = 0 for i < m ≤ k.
Thus we have to show that λi − λn 6= 0 ∀i < n as n → ∞. For small values
75
II
Differential equations in the complex plane
of i (for example, i < k) we have λi < ∞ and λn → ∞ as n → ∞, implying
λi − λn 6= 0. For larger values of i (as 0 < m < k ≤ i) we get
k
X
k
X
n!
i!
−
qm,m
(n
−
m)!
(i
−
m)!
m=0
m=0
k
X
n!
i!
.
=
qm,m
−
(n − m)! (i − m)!
m=0
λn − λ i =
qm,m
Dividing the last expression by
n!
(n−k)!
−
i!
(i−k)!
we obtain
k−1
X
λn − λ i
=
q
+
qm,m
k,k
i!
n!
(n−k)! − (i−k)!
m=1
n!
(n−m)!
n!
(n−k)!
−
−
i!
(i−m)!
i!
(i−k)!
which tends to qk,k = 1 as n → ∞. Therefore λn − λi will be nonzero. This can
be shown by considering the quotient
(n−m)! i!
i!
n!
n!
(n−m)! − (i−m)!
(n−m)! i! − (i−m)!
= lim
lim
i!
n!
(n−k)! n!
i!
n→∞
n→∞
(n−k)! − (i−k)!
(n−k)! i! − (i−k)!
(n−m)! n!
(n − k)!
i! − (i−m)!
= lim
·
(n−k)! n!
n→∞ (n − m)!
i! − (i−k)!
=0
since m ≤ k − 1. Thus, as n → ∞, every diagonal element of M becomes
non-zero and so its determinant will be non-zero, implying that M is invertible.
Thus the system will have a unique solution X = M −1 Y , where the vector X
represents the monic eigenpolynomial pn .
Remark. If deg Qm = m for at least one m (not necessarily k) and if the
coefficients qm,m of all such Qm have equal sign, then there exists a unique
monic eigenpolynomial of degree n for every value of n. To show this consider
as before the determinant of the matrix
n−1
X
Y
i!
qm,m ·
det M =
− λn .
(i − m)!
i=0
0≤m≤min(i,k)
For i ≥ k the i:th factor of this product equals
X
i!
n!
qm,m
.
−
(i − m)! (n − m)!
0≤m≤k
76
II
Differential equations in the complex plane
This expression is non-zero since i < n, and by assumption all the qm,m have
equal sign and qm,m 6= 0 for at least one m. For i < k the i:th factor equals
X
X
i!
n!
n!
−
qm,m
−
=
qm,m ·
(i − m)! (n − m)!
(n − m)!
0≤m≤i
=−
X
0≤m≤i
i+1≤m≤k
qm,m
n!
i!
−
−
(n − m)! (i − m)!
X
i+1≤m≤k
qm,m ·
n!
.
(n − m)!
This is also non-zero, since all terms have equal sign and at least one term is
non-zero. Thus every factor in the product defining the determinant is non-zero
and we get a unique solution of M X = Y for every value of n.
2.3
Probability measures whose Cauchy transform satisfies an algebraic equation
In this section we will prove the uniqueness part of Theorem 2 and show that
the measure µQk , if it exists, has the properties stated in Theorem 3. The proof
relies heavily on the following lemma.
Lemma 2. Let A ⊂ C be a finite set, U ⊂ C a convex domain and χ : U → A
a measurable function such that ∂χ/∂ z̄ ≥ 0 (in the sense of distributions). Let
a ∈ A, z0 ∈ U and assume that χ−1 (a) ∩ {|z − z0 | < r} has positive Lebesgue
measure for every r > 0. Then χ(z) = a almost everywhere in U ∩ (z 0 + Γa )
where
Γa = {z ∈ C; Re(az) ≥ Re(bz), ∀b ∈ A}.
(11)
Note that if χ−1 (a) ∩ {|z − z0 | < r} has positive Lebesgue measure for every
a ∈ A and all r > 0, then χ is determined completely (outside a set of measure
0) since the cones Γa cover the whole complex plane.
Proof. Let χa denote the characteristic function of the set χ−1 (a). We will
show that if z1 , z2 ∈ U with z2 − z1 ∈ Γa , and φ is a positive test function such
that z1 + supp φ and z2 + supp φ are both contained in U , then
(φ ∗ χa )(z1 ) ≤ (φ ∗ χa )(z2 ).
(12)
The desired conclusion follows from this. Indeed,
let φj be a sequence of positive
R
test functions such that supp φj → 0 and φj dλ = 1, where λ denotes planar
Lebesgue measure. We know then that φj ∗ χa converges in L1loc to χa . Hence,
for any , r > 0 we can find for all sufficiently large j a point z1 with |z1 −z0 | < r
such that (φj ∗ χa )(z1 ) > 1 − . It follows from (12) that (φj ∗ χa )(z2 ) > 1 − and hence
Z
|(φj ∗ χ)(z2 ) − a| = φj (z2 − ζ)(χ(ζ) − a) dλ(ζ) < max |b − a|
b∈A
77
II
Differential equations in the complex plane
for all z2 ∈ z1 + Γa . Letting and r tend to 0 and j → ∞ it follows that
χ(z) = limj→∞ (φj ∗ χ)(z) = a for almost all z in z0 + Γa .
We now prove the inequality (12). Without loss of generality we may assume
that z2 − z1 > 0 and that a = 0, for the general case can be reduced to this
situation by replacing χ with the function eiθ (χ(eiθ z)−a) where θ = arg(z2 −z1 ).
The assumption that z2 − z1 ∈ Γa then implies that A is contained in the closed
left half plane {Re z ≤ 0}.
For any > 0, let χ̃ = log(χ − ) where we have chosen a branch of the
logarithm function which is continuous in the left half plane. Let ψ be a positive
test function and note that ∂(ψ ∗ χ)/∂ z̄ ≥ 0 and Re ψ ∗ χ ≤ 0. It follows that
1
∂(ψ ∗ χ)
∂
log(ψ ∗ χ − ) = Re
·
Re
≤ 0.
∂ z̄
ψ∗χ−
∂ z̄
R
When supp ψ → 0 with ψ dλ = 1, we have that log(ψ ∗ χ − ) → χ̃ in L1loc ,
and hence as a distribution. By passing to the limit it follows that
∂ χ̃
≤ 0.
∂ z̄
If we write χ̃ = σ + iτ , this means that
Re
∂τ
∂σ
≤
.
∂x
∂y
(13)
Fix a positive test function φ such that zj + supp φ ⊂ U for j = 1, 2 and
consider the function (φ ∗ σ )(z1 + ξ) of the real variable ξ. It follows from (13)
and the fact that τ is uniformly bounded for all that
Z
∂
∂φ
(φ ∗ σ )(z1 + ξ) =
(z1 + ξ − ζ)σ (ζ) dλ(ζ)
∂ξ
∂x
Z
∂φ
≤
(z1 + ξ − ζ)τ (ζ) dλ(ζ)
∂y
≤M
where the constant M does not depend on . In particular,
(φ ∗ σ )(z2 ) − (φ ∗ σ )(z1 ) ≤ M |z2 − z1 |.
(14)
On the other hand it is clear that
(φ ∗ σ )(z) = log · (φ ∗ χa )(z) + O(1).
Now (12) follows from (14) and (15) when → 0.
(15)
We deduce two corollaries of Lemma 2.
Corollary 1. Let U ⊂ C be a convex domain and A ⊂ C a finite set. If v is
a subharmonic function defined in U such that 2∂v/∂z ∈ A almost everywhere,
then v is convex.
78
II
Differential equations in the complex plane
Recall that a subharmonic function can locally be written as the sum of a
harmonic function and a logarithmic potential. It follows that the distribution
∂v/∂z is represented by a locally integrable function. The condition 2∂v/∂z ∈ A
should be interpreted by saying that 2∂v/∂z is represented by a measurable
function with values in A.
Proof. Let χ = 2∂v/∂z. Since v is subharmonic, ∂χ/∂ z̄ ≥ 0. Take any point
z0 ∈ U and let A0 be the set of all a ∈ A such that χ−1 (a) has positive measure
in every neighborhood of z0 . Let U0 be a convex neighborhood of z0 such that
χ(z) ∈ A0 almost everywhere in U0 . By Lemma 2, χ(z) = a almost everywhere
in U0 ∩ (z0 + Γa ) where Γa is defined by (11) but with A0 in place of A. This
implies that v(z) = v(z0 ) + Re a(z − z0 ) for all z ∈ U0 ∩ (z0 + Γa ), so that
v(z) = v(z0 ) + max Re a(z − z0 ),
a∈A0
z ∈ U0 .
We have shown that in a neighborhood of z0 , v is the maximum of certain linear
functions, hence it is convex there. Since z0 was arbitrary, it follows that v is
convex.
Corollary 2. Let A ⊂ C be a finite set, U ⊂ C a convex domain and let
χ : U → A be a measurable function. Then ∂χ/∂ z̄ ≥ 0 if and only if there exist
real numbers ca (possibly equal to −∞) such that χ(z) = a almost everywhere
in Ga where
Ga = {z ∈ U ; ca + Re(az) ≥ cb + Re(bz), ∀b ∈ A}.
Proof. Suppose ca are real numbers such that χ(z) = a almost everywhere in
Ga . Let v(z) = maxa∈A (ca + Re(az)). Then v is subharmonic and χ = 2∂v/∂z,
hence
∂χ
∂2v
=2
≥ 0.
∂ z̄
∂z∂ z̄
Suppose conversely that ∂χ/∂ z̄ ≥ 0. Since ∂χ/∂ z̄ is real, there exists a real
valued function v defined in U with 2∂v/∂z = χ. It follows from Corollary 1
that v is convex. Moreover, we see from the proof that
v(z) = max (ca + Re(az))
a∈A
where
ca = inf (v(z) − Re(az)).
z∈U
If we define Ga using these constants ca it follows that v(z) = ca + Re(az) for
z ∈ Ga , hence χ(z) = 2∂v/∂z = a in Ga .
79
II
Differential equations in the complex plane
Fix a monic polynomial Qk of degree k and suppose that µ is a compactly
supported probability measure whose Cauchy transform C(z) satisfies
C(z)k = 1/Qk (z).
(16)
We will first show that µ has the properties asserted in Theorem 3, except that
supp µQk is contained in the convex hull of the zeros of Qk , which will be proved
in section 2.5.
Lemma 3. If the Cauchy transform of µ satisfies (16), then the support of µ
is the union of finitely many smooth curve segments. These curves are mapped
to lines by Ψ.
Proof. It is sufficient to prove that supp µ has these properties in a neighborhood of any given point z0 . Assume first that Qk (z0 ) 6= 0. Choose a branch
of Qk (z)−1/k defined in a simply connected neighborhood of z0 and let Ψ be a
primitive function of Qk (z)−1/k . Let U be a convex neighborhood of Ψ(z0 ) so
small that Ψ maps a neighborhood of z0 bijectively onto U . By (16) we can
write C(z) = χ(Ψ(z))Qk (z)−1/k for z ∈ Ψ−1 (U ), where χ has values in the set
of kth roots of unity. If we write w = Ψ(z), then
∂C
∂Ψ
∂χ(Ψ(z)) −1/k
∂χ
∂χ
−1/k
πµ =
·
· |Qk |−2/k
=
· Qk
= Ψ∗
· Qk
= Ψ∗
∂ z̄
∂ z̄
∂ w̄
∂z
∂ w̄
where Ψ∗ denotes the pullback of distributions in U by Ψ. Since µ is positive,
it follows that
∂χ
≥ 0.
∂ w̄
By Corollary 2, U is the union of sets Ga whose boundaries are finite unions
of line segments, such that χ is constant in each Ga . It follows that supp µ ∩
Ψ−1 (U ) = Ψ−1 (supp ∂χ/∂ z̄) is the union of finitely many curve segments which
are mapped to straight lines by Ψ.
If z0 is a zero of Qk , we take a disc D centered at z0 which does not contain
any other zeros of Qk . If γ is any ray emanating at z0 , we can define single
valued branches of Q(z)−1/k and Ψ in D r γ. Notice that Ψ is continuous up
to z0 . Let U be any half disc centered at Ψ(z0 ) and contained in Ψ(D r γ). It
follows as in the first part of the proof that supp µ has the required properties in
Ψ−1 (U ). By varying γ and U , we see that the same holds in a full neighborhood
of z0 .
Hence supp µ can be thought of as a graph whose edges are smooth curve
segments connecting certain vertices. The statement that supp µ is connected
and has connected complement then means that it is a connected graph without
cycles, that is a tree. Recall that a connected graph is a tree precisely if the
number of vertices exceeds the number of edges by exactly one.
80
Differential equations in the complex plane
II
Lemma 4. If the Cauchy transform of µ satisfies (16), then the support of µ
is a tree.
Proof. We will first prove that supp µ is connected. To do this we will show
that if U is a bounded domain which is connected and simply connected, and the
boundary of U does not intersect supp µ, then either supp µ ⊂ U or supp µ ⊂
C r U . From this it easily follows that supp µ is connected. Now it is clear that
all the zeros of Qk are either contained in U or in the complement of U , since
C(z) defines a continuous branch of Qk (z)−1/k along ∂U . Observe also that
Z
Z
Z Z
1
dz
1
dµ(ζ).
(17)
C(z) dz =
dµ(ζ) =
2πi ∂U
2πi C ∂U z − ζ
U
Now if all the zeros of Qk are contained in the complement of U , there is an
analytic continuation of C(z) across U , hence the left hand side of (17) vanishes.
It follows that supp µ ⊂ C r U . If on the other hand, all the zeros of Qk are
contained in U , then C(z) has an analytic continuation in C r U which is
asymptotically equal to a/z for some kth root of unity a when z → ∞. Thus
the left hand side of (17) is equal to a. Since the right hand side is positive, a
must be 1, which means that all the mass of µ is in U . Hence we have proved
that supp µ is connected.
Now let E be the set of all curve segments in supp µ and let V be the set of
vertices which are endpoints of the edges in E. We may assume that V contains
all the zeros of Qk . To every pair e ∈ E, v ∈ V such that v is an endpoint
of e, we assign a number ν(e, v) by the following rule. Let γ be a small loop
winding once around v in the clockwise direction, and let ν(e, v) be the jump
of (2πi)−1 log C(z) when z crosses e moving along γ. This number, which is
defined modulo Z, will be uniquely determined if we require that 0 < ν(e, v) < 1.
Assume now that v is not a zero of Qk and let e1 , . . . , er be the curves in E having
v as one endpoint. (If some curve has both its endpoints in v, it will be counted
twice.) Select a branch of Qk (z)1/k near v and observe that by Lemma 2 and the
proof of Lemma 3, Qk (z)1/k C(z) is a kth root of unity, which moves once around
the unit circle in the counterclockwise direction as z moves along γ. It follows
that ν(e1 , v) + · · · + ν(er , v) = 1. If instead v is a zero of Qk of multiplicity m, a
slight modification of the argument shows that ν(e1 , v)+· · ·+ν(er , v) = 1−m/k.
On the other hand, it is clear that ν(e, v1 ) + ν(e, v2 ) = 1 where v1 , v2 are the
endpoints of e ∈ E. Hence the sum of all the ν(e, v) is equal both to ]V − 1 and
to ]E. Since supp µ is a connected graph, this implies that it is a tree.
We are now ready to prove the uniqueness part of Theorem 2. This is done
by means of the following two lemmas.
Lemma 5. Suppose the Cauchy transform of µ satisfies (16) and let u be the
logarithmic potential of µ. If Ψ−1 is a (locally defined) inverse of a primitive
function of Qk (z)−1/k , then u ◦ Ψ−1 is convex.
81
II
Differential equations in the complex plane
Proof. Let χ be as in the proof of Lemma 3. Since 2∂u/∂z = C(z) we have
2
∂
∂u
u(Ψ−1 (w)) = 2 (Ψ−1 (w)) · Qk (Ψ−1 (w))1/k
∂w
∂z
= C(Ψ−1 (w)) · Qk (Ψ−1 (w))1/k
= χ(w).
It follows from Corollary 1, that u ◦ Ψ−1 is convex.
Lemma 6. Let µ be a measure whose Cauchy transform satisfies (16), let Ω =
C r supp µ and let Ψ(z) be defined in Ω by
Z
Ψ(z) = log(z − ζ) dµ(ζ).
Then Ψ is a multivalued function mapping Ω onto a domain H = {w; Re w >
h(Im w)} where h is a continuous function, and Ψ−1 : H → Ω is a single valued
function.
Proof. It is clear that Ψ is a holomorphic function in Ω defined up to multiples
of 2πi and that Ψ0 (z) = C(z). Let γ be a curve segment of supp µ and let U
be a one-sided neighborhood of γ in Ω on which Ψ has a single valued branch.
Now the restriction of Ψ to U has an analytic continuation across γ, and by
Lemma 3, Ψ maps γ to a line segment. Moreover, since in the notation of the
proof of Lemma 3, χ = 1 in Ψ(U ) and Re χ ≤ 1 everywhere, it follows that
Ψ(γ) is not horizontal and that Ψ(U ) is on the right hand side of Ψ(γ). Putting
the segments Ψ(γ) together as U moves around supp µ, we obtain a broken line
of the form {Re w = h(Im w)} bounding a domain H = {Re w > h(Im w)}. It
is clear that Ψ maps Ω into H and the boundary of Ω to the boundary of H.
Now ψ(z) = exp(−Ψ(z)) is a single valued proper mapping from Ω ∪ {∞} to
D = {ζ; log |ζ| < −h(− arg ζ)} which does not vanish in Ω and has a simple zero
at ∞. It follows that ψ : Ω∪{∞} → D is a bijection, hence Ψ−1 (w) = ψ −1 (e−w )
is a single valued holomorphic mapping.
Corollary 3. If µ1 and µ2 are two probability measures whose Cauchy transforms satisfy (16), then µ1 = µ2 .
Proof. Let Ψ be defined as in Lemma 6 with µ1 in place of µ, and let u1 and
u2 be the logarithmic potentials of µ1 and µ2 . Then u1 (Ψ−1 (w)) = Re w for all
w ∈ H and u2 (Ψ−1 (w)) = Re w when Re w is sufficiently large. Since u2 ◦ Ψ−1
is convex by Lemma 5, it follows that u2 (Ψ−1 (w)) ≥ Re w for all w ∈ H, hence
u1 (z) ≤ u2 (z) for almost all z. Similarly, u2 (z) ≤ u1 (z) for almost all z, and it
follows that µ1 = ∆u1 /2π = ∆u2 /2π = µ2 .
82
Differential equations in the complex plane
2.4
II
Root measures and the Cauchy transform
In this section we describe the basic connections between root measures and the
Cauchy transform which will be used to prove Theorem 4.
Let µn be a sequence of measures in the complex plane. The sequence is said
to converge weakly to a measure µ if
Z
Z
φ(z) dµn (z) → φ(z) dµ(z)
for every continuous function φ with compact support. If in addition there
exists a compact set K such that supp µn ⊂ K for every n, we will say that µn
converges weakly with compact support to µ and write µn → µ (w.c.s.).
If K ⊂ C is a compact set and M (K) denotes the space of all probability
measures with support in K, equipped with the weak topology, it is known that
M (K) is a sequentially compact Hausdorff space. We will use this fact to select
a convergent subsequence from a sequence of measures as a first step in the
proof of Theorem 4.
If φ is a locally integrable function and µ is a compactly supported measure,
the convolution
Z
(φ ∗ µ)(z) = φ(z − ζ) dµ(ζ)
is a locally integrable function defined almost everywhere in the complex plane.
If µn → µ (w.c.s.), it is easy to show that φ ∗ µn → φ ∗ µ in L1loc .
We will be particularly interested in the cases where φ(z) = log |z| or φ(z) =
1/z. Convolution with these functions defines the logarithmic potential
Z
u(z) = log |z − ζ| dµ(ζ)
and the Cauchy transform
C(z) =
Z
dµ(ζ)
z−ζ
of µ. It is well known that the measure µ can be reconstructed from either u or
C by the formula
1
1 ∂C
µ=
· ∆u = ·
2π
π ∂ z̄
where ∆ = (∂/∂x)2 + (∂/∂y)2 is the Laplace operator and ∂/∂ z̄ = (∂/∂x +
i∂/∂y)/2. These identities should be understood in the sense of distribution
theory.
83
II
Differential equations in the complex plane
Let p be a polynomial of degree n and let µ be the root measure of p, as
defined in the introduction. If p is monic, the logarithmic potential of µ is given
by
Z
1
log |p(z)| = log |z − ζ| dµ(ζ),
(18)
n
and for any p, the Cauchy transform of µ is
Z
dµ(ζ)
p0 (z)
=
.
np(z)
z−ζ
(19)
These two identities, which can easily be verified, are among the main ingredients in the proof of Theorem 4. We will here use them to prove a general lemma
which will be needed later.
Lemma 7. Let pm be a sequence of polynomials, such that nm := deg pm → ∞
and let µm and µ0m be the root measures of pm and p0m respectively. If µm → µ,
µ0m → µ0 (w.c.s.) and u and u0 are the logarithmic potentials of µ and µ0 , then
u0 ≤ u with equality in the unbounded component of C r supp µ.
Proof. Assume with no loss of generality that pm are monic. Let K be a
compact set containing the zeros of every pm . By (18) we then have
u(z) = lim
m→∞
and
1
log |pm (z)|
nm
0
0
pm (z) pm (z) 1
1
log = lim
log u (z) = lim
m→∞ nm − 1
nm m→∞ nm
nm 0
with convergence in L1loc . Hence by (19),
0
Z
pm (z) dµm (ζ) 1
1
0
.
u (z) − u(z) = lim
log log = lim
m→∞ nm
nm pm (z) m→∞ nm
z−ζ Now, if φ is a positive test function it follows that
Z
Z
Z
dµm (ζ) 1
dλ(z)
φ(z) log φ(z)(u0 (z) − u(z)) dλ(z) = lim
m→∞ nm
z−ζ Z
Z
dµm (ζ)
1
φ(z)
dλ(z)
≤ lim
m→∞ nm
|z − ζ|
ZZ
φ(z) dλ(z)
1
= lim
dµm (ζ)
m→∞ nm
|z − ζ|
(20)
(21)
where λ denotes Lebesgue
R measure in the complex plane. Since 1/|z| is locally
integrable, the function φ(z)|z − ζ|−1 dλ(z) is continuous, and hence bounded
by a constant M for all z in K. Since supp µm ⊂ K, the last expression in (21)
84
Differential equations in the complex plane
II
is bounded by M/nm , hence the limit when m → ∞ is 0. This proves that
u0 ≤ u.
In the complement of supp µ, u is harmonic and u0 is subharmonic, hence
u0 − u is a negative subharmonic function. Moreover, in the complement of
K, p0m /(nm pm ) converges uniformly on compact sets to the Cauchy transform
C(z) of µ. Since C(z) is a nonconstant holomorphic function in the unbounded
component of C r K, it follows from (20) that u0 − u = 0 there. By the
maximum principle for subharmonic functions it follows then that u0 − u = 0 in
the unbounded component of C r supp µ. The proof is complete.
2.5
Root measures of eigenpolynomials
We now turn to the proof of Theorem 4. The plan is to show that µn converges
to a measure whose Cauchy transform satisfies (16). This will prove Theorem
4 and the existence part of Theorem 2. Let µn be the root measure of pn as
(j)
in the statement of Theorem 4. Also let µn be the root measure of the jth
(j)
derivative pn . We begin by showing that there is a compact set K containing
(j)
the supports of all the measures µn .
Lemma 8. Let Q0 , ..., Qk be fixed and let pn be the eigenpolynomial of degree
n of the operator TQ . Then there exists a compact set K such that all the zeros
(i)
of every pn lie in K for every n and every i ≥ 0. If Q0 = . . . = Qk−1 = 0, K
may be taken as the convex hull of the zeros of Qk .
Proof. It suffices to check the roots of pn , since by Gauss-Lucas’ theorem
(i)
the roots of any derivative pn are contained in the convex hull of the roots of
pn . Furthermore it suffices to show that there exists a compact set containing
the zeros of pn for large values of n, since for any finite value of n we have
finitely many roots of the polynomial pn , and these are clearly contained in
some compact set.
Let z be a root of pn . Then
TQ (pn )(z) =
k
X
i=0
Qi (z) · p(i)
n (z) = λn · pn (z) = 0
or, equivalently,
(k−1)
Qk (z) · p(k)
(z) + . . . + Q1 (z) · p(1)
n (z) + Qk−1 (z) · pn
n (z) = 0.
(22)
We will show that for sufficiently large choices of |z| and n this equation will
not hold. It is possible to find some r0 and some n0 such that if |z| > r0 and
n > n0 then z cannot be a root of pn . Using formula (19) we have
Z
(i)
(i+1)
dµn (ζ)
pn (z)
=
=: bi .
(i)
z−ζ
(n − i) · pn (z)
85
II
Differential equations in the complex plane
Thus
(k)
pn(k−1) (z) =
(k−1)
pn(k−2) (z) =
pn (z)
,
(n − k + 1) · bk−1
(k)
pn
(z)
pn (z)
=
,
(n − k + 2) · bk−2
(n − k + 1)(n − k + 2) · bk−1 · bk−2
and so on. Generally
(k)
p(i)
n (z) =
pn
(n − k + 1) . . . (n − i) ·
Qk−1
j=i
bj
.
Now assume that z is the root of pn with the largest modulus and let |z| = r.
(i)
With ζ being a root of some pn we have |ζ| ≤ |z| by Gauss-Lucas’ theorem.
R dµ(i)
n (ζ)
so that |bi | ≥ 1/2r ∀i ≤ k.
We will estimate bi =
z−ζ
We have
1
1
1
1
1
= ·
= ·
z−ζ
z 1 − ζ/z
z 1−θ
and |θ| = |ζ/z| ≤ 1. With w = 1/(1 − θ) we obtain
1
|θ|
(1 − θ) |w − 1| = =
−
= |θ||w| ≤ |w|
1 − θ (1 − θ) |1 − θ|
⇔
|w − 1| ≤ |w|
m
1
Re(w) ≥ .
2
Using this result we get
Z
dµ(i) (ζ) 1 Z dµ(i) (ζ) n
n
|bi | = = z−ζ r 1−θ Z
Z
1
1 (i)
(i)
= wdµn (ζ) ≥ Re(w)dµn (ζ)
r
r
Z
1
1
≥
· dµ(i)
.
n (ζ) =
2r
2r
Now we choose r0 in such a way that |Qk (w)| ≥ rk /2 as |w| ≥ r0 and then a
constant C such that |Qi (w)| ≤ C · ri for every i = 1, . . . , k − 1. Finally we
C·2k−i+1
1
choose n0 such that (n−i)...(n−k+1)
< k−1
as n > n0 for every i = 1, . . . , k − 1.
86
II
Differential equations in the complex plane
Then as |z| = r ≥ r0 and n > n0 we have
Q (z) · p(i) (z) 1
|Qi (z)| (n − k)!
n
i
·
· Qk−1
=
Qk (z) · p(k)
|Q
(z)|
(n
−
i)!
k
n (z)
j=i |bj |
|Qi (z)| (n − k)! k−i k−i
·
·2
·r
|Qk (z)| (n − i)!
C · ri (n − k)! k−i k−i
·r
·
·2
≤ k
r /2 (n − i)!
≤
=
1
C · 2k−i+1
<
.
(n − i) . . . (n − k + 1)
k−1
(k)
Dividing (22) by Qk (z) · pn (z) we obtain
1+
k−1
X
i=1
(i)
Qi (z) · pn (z)
(k)
Qk (z) · pn (z)
= 0,
but with r ≥ r0 and n > n0 we get
k−1
k−1
(i)
X Q (z) · p(i) (z) k−1
n
X Qi (z) · pn (z) X 1
i
=1
≤
<
(k)
(k)
k−1
i=1 Qk (z) · pn (z)
i=1 Qk (z) · pn (z)
i=1
and so (22) cannot be fulfilled with such choices of r and n.
Assume that N is a subsequence of the natural numbers such that
µ(j) =
lim
n→∞,n∈N
µ(j)
n
(23)
exists for j = 0, . . . , k. The following lemma shows that the Cauchy transform
of µ = µ(0) satisfies (16).
Lemma 9. The measures µ(j) are all equal and the Cauchy transform C(z) of
this common limit satisfies C(z)k = 1/Qk (z) for almost every z.
Proof. By (19) we have that
(j+1)
pn
(z)
(j)
(n − j)pn (z)
→
Z
dµ(j) (ζ)
z−ζ
(24)
with convergence in L1loc , and by passing to a subsequence once again we can assume that we have pointwise convergence almost everywhere. From the relation
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II
Differential equations in the complex plane
TQ pn = λn pn it follows that
(k)
Qk
pn
λn
=
n . . . (n − k + 1)pn
n . . . (n − k + 1)
−
k−1
X
l=0
l−1
Y p(j+1)
Ql
n
.
(n − l) . . . (n − k + 1) j=0 (n − j)p(j)
n
(25)
Now λn /n . . . (n − k + 1) → 1 by Theorem 1, while the sum converges pointwise
to 0 almost everywhere by virtue of the factors (n − l) . . . (n − k + 1) in the
denominators. It follows that
(k)
1
pn (z)
→
n . . . (n − k + 1)pn (z)
Qk (z)
(26)
when n → ∞ through the sequence N for almost every z. If u(j) denotes the
logarithmic potential of µ(j) , then it follows from (18) and (26) that
(k)
p
1
1
n
u(k) − u(0) = lim
log log |Qk | = 0.
= − lim
n→∞ n
n→∞ n
n . . . (n − k + 1)pn On the other hand we have from Lemma 7 that u(0) ≥ u(1) ≥ · · · ≥ u(k) , hence
the potentials u(j) are all equal, and it follows that µ(j) = ∆u(j) /2π are all
equal. Finally we have from (24) and (26) that
C(z)k = lim
n→∞
k−1
Y
j=0
(j+1)
pn
(z)
(j)
(n − j)pn (z)
(k)
pn (z)
1
=
n→∞ n . . . (n − k + 1)pn (z)
Qk (z)
= lim
for almost every z. This completes the proof.
Corollary 4. There exists a unique measure µQk satisfying the requirements in
Theorem 2. The sequence µn converges weakly to µQk . Moreover, supp µQk is
contained in the convex hull of the zeros of Qk .
Proof. By Theorem 1, the operator TQ has an eigenpolynomial pn of degree n
for all sufficiently large n. By Lemma 8, there exists a compact set K such that
(j)
supp µn ⊂ K for all n. By compactness, there exists a subsequence N such that
the limit (23) exists for j = 0, . . . , k. By Lemma 9, µQk = µ(0) has the required
properties, so existence is proved. Uniqueness was established in section 2.3.
(j)
Since we may take Q0 = . . . = Qk−1 = 0, and in this case supp µn ⊂ K where
K is the convex hull of the zeros of Qk by Lemma 8, it follows that supp µQk is
also contained in K.
Assume that µn does not converge to µQk Then we can find a subsequence
N 0 of the natural numbers such that µn stays away from a fixed neighborhood
88
Differential equations in the complex plane
II
of µQk in the weak topology, for all n ∈ N 0 . Again by compactness, we can
find a subsequence N of N 0 such that the limit (23) exists for j = 0, . . . , k. By
Lemma 9 and the uniqueness of µQk , it follows that µ(0) = µQk , contradicting
the assumption that µn stays away from µQk for all n in N 0 and hence all n in
N . The proof is complete.
References
[1] K. Kwon, L. Littlejohn, G. Yoon: Bochner-Krall orthogonal polynomials,
p.181–193 in Special functions, World Sci. Publishing, 2000.
[2] L. Hörmander: The analysis of partial differential operators I, SpringerVerlag.
[3] E. Kamke: Differentialgleichungen, Lösungsmetoden und Lösungen, Becker
& Erler, Leipzig, 1942.
[4] G. Masson, B. Shapiro: On polynomial eigenfunctions of a hypergeometrictype operator, Experiment. Math. 10 (2001), no. 4, 609–618.
[5] T. Ransford: Potential theory in the complex plane, Cambridge University
Press, 1995.
[6] H. Shapiro: Spectral aspects of a class of differential operators, in Operator
methods in ordinary and partial differential equations (Stockholm, 2000),
Birkhäuser, 2002, 361–385.
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Radon transforms and
tomography
III
Radon transforms and tomography
1
Introduction
Consider a function f defined on the n-dimensional Euclidean space Rn . The
Radon transform of f is the function defined on the set of all hyperplanes in
Rn , which assigns to every hyperplane the integral of f over that hyperplane.
The Radon transform was first considered by Johann Radon in 1917, who asked
if the function f is uniquely determined by its Radon transform, and answered
the question by giving an explicit formula for computing f .
The Radon transform can be generalized in several ways. In one direction,
the hyperplanes can be replaced by subspaces of some other fixed dimension.
When the function is integrated over lines, the transform is known as an Xray transform. Another generalization is to multiply f by a weight function,
depending on the subspace, before the integration is carried out. In this case
we speak of weighted Radon or X-ray transforms.
Transforms of this type arise in different kinds of tomography. The general
problem in tomography is to determine, non-destructively, the three-dimensional
structure of a physical object. This can be accomplished by measurements on
radiation passing through the object, for example X-rays or electron beams from
external sources or gamma rays from radioactive material inside the object.
If the object is represented by a function f on three-dimensional space, the
measured data typically represent a sampling of some form of X-ray transform
of f . Therefore it becomes necessary to compute f given some measurements
of the relevant X-ray transform. In certain cases, the problem can be reduced
to a consideration of two-dimensional Radon transforms, applied to slices of the
object.
This chapter is organized as follows. In Section 2 we give definitions of the
Radon transforms that will be considered, outline some important problems
related to these and give a brief overview of some known results. Section 3
contains various definitions and lemmas of a rather general nature which will
be used later on. Section 4 deals with forward estimates, which are statements
to the effect that the weighted Radon transform of f depends continuously on
f and the weight function. In Section 5 we obtain some new stability results
for the inverse problem, which means that f can be stably determined if its
weighted Radon transform and the weight function are known approximately.
Finally, in Section 6 we give a new inversion formula for the exponential Radon
transform. The two main results are Corollary 2 in Section 5 and Corollary 6
in Section 6.
2
Background
In this section we define the Radon transforms with which we will be concerned,
list some important problems related to these, and give references to some known
results.
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III
2.1
Radon transforms and tomography
Definitions
Let f be a function defined in the plane R2 . By a weight function we mean a
function ρ(θ, x) defined on S 1 × R2 . The weighted Radon transform of f with
weight ρ is the function Rρ f defined on S 1 × R by the integral
Z
Rρ f (θ, s) =
ρ(θ, x)f (x) dl.
(1)
x·θ=s
The integration takes place along a line perpendicular to θ, displaced a distance s
from the origin, and dl denotes linear measure on this line. We use the notation
R for the (unweighted) Radon transform with weight ρ ≡ 1. If the weight
function has the property that ρ(θ, x) = ρ(−θ, x) for almost all (θ, x) ∈ S 1 × R,
we say that it is even. If ρ is an even weight function and f is arbitrary, the
Radon transform g = Rρ f satisfies g(θ, s) = g(−θ, −s), and we say that g is
even if it has this property.
The attenuated Radon transforms are of particular interest in applications
and are defined by weight functions of the following form. Let µ be a compactly
supported function defined in R2 and let
Z ∞
⊥
µ(x + tθ ) dt .
(2)
ρ(θ, x) = exp −
0
⊥
Here θ denotes the vector obtained by rotating θ counter-clockwise through a
right angle. We call µ an attenuation function and say that ρ is of attenuation
type when it is of this form. We use the notation Rµ for the attenuated Radon
transform.
A still more restrictive class of Radon transforms which have been studied
in detail are the exponential Radon transforms, defined by weight functions of
the form
ρ(θ, x) = exp(µx · θ ⊥ )
(3)
where µ is a real constant called the attenuation. We denote the exponential
Radon transform also by Rµ . These are of interest for the following reason.
Suppose f is a compactly supported function, and that µ is an attenuation
function which is equal to a constant µ0 in the convex hull of supp f . Then it
is easy to see that
Z ∞
Rµ f (θ, s) = Rµ0 f (θ, s) exp −
µ(sθ + tθ⊥ ) dt ,
(4)
0
so the attenuated Radon transform Rµ f can be expressed in terms of the much
simpler exponential Radon transform Rµ0 f .
2.2
Important problems
The practical applications of Radon transforms in general involve solving an
inverse problem, that is, recovering the function f from measurements of Rρ f .
As in any inverse problem, the following considerations are essential.
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Radon transforms and tomography
• Injectivity. Is Rρ injective, so that f is uniquely determined by Rρ f ?
• Stability. In general, it is only possible to have an approximation of Rρ f
sampled at a discrete set of points, so it is of interest to know how good
an approximation of f can be obtained from such data. A stability result
for the inverse problem is an estimate of the form
kf k(1) ≤ CkRρ f k(2)
(5)
for some constant C not depending on f , and some function norms k · k(1)
and k · k(2) .
• Inversion formula. Can one find an explicit formula expressing f in
terms of Rρ f for some class of weight functions ρ?
All of these problems can be considered under various restrictions on the
available data. In limited angle tomography the Radon transform Rρ f (θ, s) is
only known for a limited range of angles θ. In medical applications, such as
SPECT, the available range of angles is typically a half circle, and in other
cases, for example in electron microscopy, the range of angles is even smaller.
In local tomography one seeks to reconstruct important features of the function f in a region D using only integrals along lines intersecting D, and the
goal of exterior tomography is to recover f in the complement of D using only
integrals along lines not intersecting D.
The so-called identification problem aims at determining both f and ρ at
once from measurements of Rρ f . For arbitrary weight functions this is of course
impossible, but for weights of attenuation type some progress has been made,
although the problem is ill posed and in general does not have a unique solution
(see [3]).
2.3
Known results
Here we give a brief overview of known results related to the problems listed in
the previous section.
Explicit formulas. An explicit inversion formula for the unweighted Radon
transform was first derived by J. Radon in 1917, see [11], and a generalization
to the exponential Radon transform was derived by Tretiak and Metz in [13].
An explicit inversion formula for the attenuated Radon transform was recently
found by Novikov, see [10]. These are the cases for which explicit inversion
formulas exist, and they all require the Radon transform to be known on all
of S 1 × R. However, in [9], an iterative algorithm is given for reconstructing
a function from its exponential Radon transform restricted to a 180◦ range of
angles.
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Radon transforms and tomography
Injectivity. It is easy to show that a compactly supported function f is
uniquely determined by its unweighted or exponential Radon transform restricted to an arbitrarily small open range of angles. This is because the
Radon transform determines the Fourier transform of f on a subset of C2 ,
which uniquely determines the Fourier transform on all of C2 by analytic continuation. A corresponding result for the attenuated Radon transform has been
obtained by Novikov in [10] and independently by Boman and Strömberg in [2].
Since analytic continuation is not numerically stable, these considerations do
not lead to useful inversion formulas.
The general weighted Radon transform need not be injective, even when the
weight function is smooth and the object function has compact support, as is
shown by the counterexample in [1].
Stability. Stability results for the unweighted Radon transform are discussed
in [7]. When an explicit inversion formula exists, it is in general possible to
deduce stability results. Moreover, Markoe and Quinto have obtained a local
invertibility and stability result for arbitrary weight functions of class C 2 (see
[6]). By this it is understood that the Radon transform uniquely and stably
determines the object function, provided that it has sufficiently small support,
depending on the weight function.
3
Preliminaries
In this section we collect some definitions and lemmas of a rather general nature
which will be needed in later sections.
3.1
Function spaces and norms
We let L2 (Rn ) denote the space of complex valued, square-integrable functions
on Rn , and kf k the L2 norm of a function f . More generally, if w is a positive
function on Rn , we let
Z
1/2
2
kf kw =
|f (x)| w(x) dx
(6)
Rn
L2w (Rn )
and let
denote the space of functions f on Rn such that kf kw is finite.
If 1 ≤ p ≤ ∞ we write kf kLp for the Lp norm of f .
For a function f defined on Rn , we let
Z
ˆ
f (ξ) =
e−ix·ξ f (x) dx
(7)
Rn
denote its Fourier transform. For any real number k, define the Sobolev norm
Z
1/2
1
2
2 k
ˆ
(8)
|f (ξ)| (1 + |ξ| ) dξ
kf kk =
(2π)n/2
Rn
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Radon transforms and tomography
and denote by H k (Rn ) the completion of the space of all functions f such that
kf kk is finite. (For k < 0, elements of H k may be distributions rather than
functions.) Note that kf k0 is the same as the L2 norm of f .
We let h·, ·i, h·, ·iw and h·, ·ik denote the scalar products corresponding to
the norms k · k, k · kw and k · kk . For an operator T on some function space, we
let T ∗ denote the adjoint of T with respect to the scalar product h·, ·i.
For any positive real number p, let Ip denote the operator defined by
(Ip f ˆ) (ξ) = |ξ|p fˆ(ξ).
(9)
This makes sense whenever the Fourier transform of f is a locally integrable
function. Note that Ip is a bounded operator from H k+p to H k for every k.
For a function f (θ, x) defined on S 1 × Rn , we define fˆ(θ, ξ) to be the Fourier
transform of f with respect to the x variable, and we define Sobolev norms of
such functions by
Z
1/2
kf kk =
kf (θ, ·)k2k dθ
(10)
S1
where dθ denotes arclength measure on S 1 . We also extend the operators Ip to
ˆ ξ).
such functions by letting (Ip f ˆ) (θ, ξ) = |ξ|p f(θ,
n
For 0 < α ≤ 1 and a function f on R we define the Hölder norm by
kf kC α = sup |f (x)| + sup
x
x6=y
|f (x) − f (y)|
.
|x − y|α
(11)
For a weight function ρ we define the Hölder norm by
kρkC α = sup kρ(θ, ·)kC α .
(12)
kρkCθα = sup kρ(θ, τ θ + tθ ⊥ )kC α
(13)
kρkC α⊥ = sup kρ(θ, sθ + τ θ ⊥ )kC α
(14)
θ
Moreover, we write
θ,t
and
θ
θ,s
where the C α -norms on the right hand side are taken with respect to the real
variable τ . We write Cθα and Cθα⊥ for the classes of weight functions such that
these norms are finite.
We let kT kk denote the operator norm of an operator T on H k .
The set B of all bounded linear operators on a Banach space constitutes
a Banach algebra. The subset K of all compact operators is an ideal in this
algebra, and the quotient C = B/K is known as the Calkin algebra. If T ∈ B,
the norm of the class of T in C is defined to be kT kC = inf K∈K kT − Kk, and
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Radon transforms and tomography
with this norm, C is a Banach algebra. The essential spectrum of an operator
T is the set of complex numbers λ such that T − λ Id is not invertible in the
Calkin algebra, where Id denotes the identity operator. We will write Bk for the
algebra of bounded operators on H k (R2 ) and Ck for the corresponding Calkin
algebra.
3.2
Estimates of operator norms
Here we derive a few simple estimates of the norms of certain integral operators,
which we will use in the following sections.
Lemma 1. Let w be a positive function on Rn and let T be an integral operator
on L2w (Rn ) with kernel ϕ(x, y). If
s
w(x)
r(z) = sup |ϕ(x, x + z)|
.
(15)
w(x + z)
x∈R2
then kT kw ≤
R
r(z) dz.
Proof. Let f1 and f2 be arbitrary L2w -functions. Then
Z
|hT f1 , f2 iw | = ϕ(x, y)f1 (y)f 2 (x)w(x) dx dy Z
= ϕ(x, x + z)f1 (x + z)f 2 (x)w(x) dx dz Z
p
≤ r(z)|f1 (x + z)||f2 (x)| w(x)w(x + z) dx dz
≤ krkL1 kf1 kw kf2 kw .
The following simple observation plays an important part in the proof of
Theorem 3.
Lemma 2. Let χ1 and χ2 be functions on Rn with |χ1 | + |χ2 | ≤ 1, let β1 and
β2 be bounded functions on Rn and let S be the operator on L2 (Rn ) defined by
(Sf ˆ) =
2
2
X
1 X
ˆ =
χ
·
(
β̂
∗
f)
χj · (βj · f ˆ) .
j
j
(2π)n j=1
j=1
(16)
Then the operator norm of S satisfies the estimate
kSk ≤
1
(kβ1 + β2 kL∞ + kβ1 − β2 kL∞ ).
2
98
(17)
III
Radon transforms and tomography
Proof. Consider the operators S± defined by (S± f ˆ) = (χ1 ± χ2 )fˆ. Then
kS± f k ≤ kf k, and it follows that
1
kS+ (β1 + β2 )f + S− (β1 − β2 )f k
2
1
≤ (kβ1 + β2 kL∞ + kβ1 − β2 kL∞ )kf k.
2
kSf k =
Lemma 3. Let ϕ(ξ, η) be a function on Rn × Rn and define for any positive
numbers A and B an operator T by
Z
ˆ
(T f ) (ξ) =
ϕ(ξ, η)fˆ(η) dη.
|ξ−η|≤A
|η|>B
Then kT kk /kT k → 1 when B → ∞ with A fixed for any real number k.
Proof. Note that kT k and kT kk are decreasing functions of B when A is
fixed. Assuming that kT k < ∞ for some B, we show that kT kk /kT k < 1 + for sufficiently large B, where is an arbitrary positive number. This shows
that the limit is less than or equal to 1 (which is what we will actually use).
The reverse inequality can be proved by the same method. We also assume, for
simplicity, that k < 0.
Let C be a positive number, and let for any r > 0,
1/C, if r − C ≤ t ≤ r
χr (t) =
0,
otherwise
and
χ̃r (t) =
1, if r − C − A ≤ t ≤ r + A
.
0, otherwise
Let f and f˜ be arbitrary functions and define fr and f˜r by fˆr (ξ) = χr (|ξ|)fˆ(ξ)
ˆ
ˆ
˜
and
χ̃r (|ξ|)Ef˜(ξ). Note that kfr k2k = hfr , fr ik = hfr , f ik /C and kf˜r k2k =
D fr (ξ)
E =D
f˜r , f˜r = f˜r , f˜ , hence
k
k
Z
and
Z
∞
B
∞
B
kfr k2k
1
dr =
C
kf˜r k2k dr =
Z
∞
B
Z
D
∞
B
hfr , f ik dr ≤
1
kf k2k
C
E
˜ 2.
f˜r , f˜ dr ≤ (C + 2A)kfk
k
k
99
III
Radon transforms and tomography
From these estimates it follows that
D
E Z ∞ D
E
˜
˜
dr
T fr , f
T f, f = k
k
ZB∞ D
E
˜
=
dr
T fr , fr
k
Z B∞
D
E
≤
(1 + (r − C − A)2 )k T fr , f˜r dr
B
Z ∞
(1 + (r − C − A)2 )k kfr kkf˜r k dr
≤ kT k
B
Z ∞
(1 + (r − C − A)2 )k
kfr kk kf˜r kk dr
≤ kT k
(1 + (r + A)2 )k
B
sZ
Z ∞
∞
(1 + (B − C − A)2 )k
2 dr
kf˜r k2k dr
kf
k
≤
kT
k
r k
(1 + (B + A)2 )k
B
B
r
(1 + (B − C − A)2 )k C + 2A
˜k
kT kkf kk kfk
≤
(1 + (B + A)2 )k
C
which shows that
(1 + (B − C − A)2 )k
kT kk ≤
(1 + (B + A)2 )k
r
C + 2A
kT k.
C
Now it is clear that, for any given A, C and B can be chosen so large that
kT kk ≤ (1 + )kT k. If k > 0 the same calculation holds if we exchange B + A
and B − C − A.
3.3
Radon transforms and Fourier transforms
In this section we derive some formulas involving Fourier transforms of weighted
Radon transforms which will be needed in the proofs of the main results.
Lemma 4. If f ∈ L2 (R2 ) and ρ is a weight function such that ρ(θ, ·) ∈ L2 (R2 )
for every θ ∈ S 1 , then
(Rf ˆ) (θ, σ) = fˆ(σθ)
(18)
and
Z
1
ρ̂(θ, σθ − η)fˆ(η) dη.
(2π)2 R2
If g is a function on S 1 × R, then
2π
ξ
ξ
(R∗ g)ˆ (ξ) =
ĝ
, |ξ| + ĝ − , −|ξ|
.
|ξ|
|ξ|
|ξ|
(Rρ f ˆ) (θ, σ) =
It follows that
1
(R I1 Rρˆ) (ξ) =
2π
∗
Z
ξ
ξ
fˆ(η) dη.
ρ̂
, ξ − η + ρ̂ − , ξ − η
|ξ|
|ξ|
R2
100
(19)
(20)
(21)
III
Radon transforms and tomography
Proof. Equation (18) is well known and follows immediately from the definition of the Radon and Fourier transforms, see for example [7, Theorem II.1.1].
If we write ρθ (x) = ρ(θ, x) and use the formula for the Fourier transform of a
product, it follows that
(Rρ f ˆ) (θ, σ) = (R(ρθ f ))ˆ (θ, σ)
= (ρθ f ˆ) (σθ)
Z
1
=
ρ̂θ (θ, σθ − η)fˆ(η) dη.
(2π)2 R2
This proves (19). To prove (20), let f be the function whose Fourier transform
is the right hand side of (20). Then
1 Dˆ ˆE
f, f1
(2π)2
Z
1
ξ
1
ξ
¯
=
ĝ
, |ξ| + ĝ − , −|ξ|
fˆ1 (ξ) dξ
2π R2 |ξ|
|ξ|
|ξ|
Z
¯
1
ĝ(θ, σ) + ĝ(−θ, −σ) fˆ1 (σθ) dθ dσ
=
2π S 1 ×R+
E
1 D
ĝ, (Rf1ˆ)
=
2π
= hg, Rf1 i
hf, f1 i =
for every test function f1 , and it follows from the definition of R∗ that f = R∗ g.
The relation (21) follows by combining (19) and (20).
Lemma 5. Let ρ(θ, x) and ρ̃(θ, x) be even weight functions, equal to constants
ρ0 and ρ̃0 for all x outside a compact set. Let
¯ 0 , x) − ρ0 ρ̃¯0
β(θ, θ0 , x) = ρ(θ, x)ρ̃(θ
(22)
and let T denote the operator
T =
Then
(T f ˆ) (ξ) =
1
R∗ I1 Rρ Rρ̃∗ I1 R − ρ0 ρ̃¯0 Id .
16π 2
1
(2π)2
Z
β̂
R2
ξ η
, , ξ − η fˆ(η) dη.
|ξ| |η|
(23)
(24)
Proof. Write ρ = ρ0 + ρ1 and ρ̃ = ρ̃0 + ρ̃1 . From (18) and (20) it follows that
R∗ I1 R = 4π Id, hence
T =
ρ0 ∗
1
ρ̃¯0 ∗
R I 1 R ρ1 +
(R I1 Rρ̃1 )∗ +
R∗ I1 Rρ1 (R∗ I1 Rρ̃1 )∗ .
4π
4π
16π 2
101
III
Radon transforms and tomography
Now (24) follows from (21) applied to the weight functions ρ1 and ρ̃1 together
with the relation
Z
¯1 (θ0 , ξ) + 1
¯1 (θ0 , ξ − η) dη.
β̂(θ, θ0 , ξ) = ρ̂1 (θ, ξ)ρ̃¯0 + ρ0 ρ̃ˆ
ρ̂1 (θ, η)ρ̃ˆ
(2π)2
4
Forward estimates
Here we shall derive some forward estimates of weighted Radon transforms. By
this we mean estimates of Rρ f in terms of a Sobolev norm of f and a Hölder
norm of ρ. These will be used in Section 5.2 to show that the stability resuts in
Section 5.1 are valid for a large class of weight functions. They are also useful
for estimating what consequences an error in the weight function will cause.
From a practical point of view, Theorem 2 and Corollary 1 are most interesting.
The estimate in Theorem 1 follows with little additional effort. It is likely that
similar estimates hold also for other values of k, but this seems more difficult
to prove.
Theorem 1. If 0 ≤ k < 1/2, k + 1/2 < α ≤ 1 and K ⊂ R2 is a compact set,
there exists a constant C such that
k
2
kI1/2 Rρ f kk ≤ CkρkC α kf kk
(25)
α
for all f ∈ H (R ) and all weight functions ρ(θ, x) of class C , constant for x
outside K. If also supp f ⊂ K, then
kRρ f kk+1/2 ≤ CkρkC α kf kk
(26)
(possibly with a larger constant C).
Theorem 2. If α > 1/2 and K ⊂ R2 is a compact set, there exists a constant
C such that
kI1/2 Rρ f k−1/2 ≤ CkρkC α⊥ kf k−1/2
(27)
θ
for all f ∈ H −1/2 (R2 ) and all weight functions ρ(θ, x) of class Cθα⊥ , constant
for x outside K. If also supp f ⊂ K, then
kRρ f k ≤ CkρkC α⊥ kf k−1/2
θ
(28)
(possibly with a larger constant C).
Corollary 1. If K ⊂ R2 is a compact set, there exists a constant C such that
kRµ f k ≤ C(1 + kµk∞
L )kf k−1/2
(29)
for all positive attenuation functions µ and all f ∈ H −1/2 (R2 ) with supp µ and
supp f both contained in K. Moreover, if µ1 and µ2 are positive attenuation
functions with support in K, then
kRµ1 f − Rµ2 f k ≤ Ckµ1 − µ2 kL∞ (1 + kµ1 kL∞ )kf k−1/2 .
102
(30)
III
Radon transforms and tomography
Proof. For (θ, x) ∈ S 1 × K, let
Z
ρ(θ, x) = exp −
∞
0
µ(x + tθ⊥ ) dt .
Then |ρ(θ, x)| ≤ 1 and for τ > 0,
Z
|ρ(θ, x + τ θ⊥ ) − ρ(θ, x)| = ρ(θ, x + τ θ⊥ ) 1 − exp −
τ
µ(x + tθ⊥ ) dt
0
≤ τ kµkL∞ .
Hence ρ can be extended to a weight function on S 1 × R2 such that kρkC 1⊥ ≤
θ
1 + kµkL∞ . Since Rµ f = Rρ f for all f with support in K, the estimate (29)
follows from Theorem 2. To prove the second estimate, let ρ1 and ρ2 be defined
like ρ in the first part, with µ1 and µ2 in the place of µ, and let ρ = ρ1 − ρ2 .
Then |ρ(θ, x)| ≤ 1 and for τ > 0,
Z τ
⊥
⊥
⊥
µ1 (x + tθ ) dt
|ρ(θ, x + τ θ ) − ρ(θ, x)| = ρ1 (θ, x + τ θ ) 1 − exp −
0
Z τ
⊥
⊥
µ2 (x + tθ ) dt − ρ2 (θ, x + τ θ ) 1 − exp −
0
≤ |ρ1 (θ, x + τ θ⊥ ) − ρ2 (θ, x + τ θ⊥ )|
Z τ
µ1 (x + tθ⊥ ) dt
1 − exp −
0
⊥
+ ρ2 (θ, x + τ θ )
Z
Z τ
exp −
µ
(t)
dt
−
exp
−
1
0
τ
0
µ2 (t) dt ≤ C1 kµ1 − µ2 kL∞ τ kµ1 kL∞ + τ kµ1 − µ2 kL∞
for some constant C1 depending only on K. Extend ρ to a weight function on
S 1 × R2 with kρkC 1⊥ ≤ Ckµ1 − µ2 kL∞ (1 + kµ1 kL∞ ). Then the estimate (30)
θ
follows from Theorem 2 since Rµ1 f − Rµ2 f = Rρ f for all f with support in K.
Remark. It follows from the proof, by using Hölder’s inequality, that the L∞
norms of the weight functions may be replaced by the norm
µ 7→ sup
θ,s
Z
⊥
R
p
|µ(sθ + tθ )| dt
for any p > 2.
103
1/p
III
Radon transforms and tomography
In the proofs we will use the following notation. For any θ ∈ S 1 , define
operators Jk and Jkθ by
and
ˆ
(Jk f ˆ) (ξ) = (1 + |ξ|2 )k/2 f(ξ)
(31)
(Jkθ f ˆ) (ξ) = (1 + |ξ · θ|2 )k/2 fˆ(ξ).
(32)
1
For a function g(θ, s) defined on S × R we define Jk to act on the second
variable. For a weight function ρ we write
ρθ (x) = ρ(θ, x)
and we write
Rρθ f (s) = Rρ f (θ, s).
Lemma 6. Let 0 < k < α ≤ 1. Then there exists a constant C such that
kJk (ρf ) − ρJk f k ≤ CkρkC α kf k
(33)
for all functions ρ ∈ C α (R) and f ∈ L2 (R).
Proof. Let φ be the distribution on R whose Fourier transform is the function
σ 7→ (1 + σ 2 )k/2 . By comparing this with the homogeneous function σ 7→ |σ|k ,
we see that φ is a locally integrable function outside the origin and that φ(s) =
O(|s|−k−1 ). Letting g = Jk (ρf ) − ρJk f , it follows that
Z
g(s) =
φ(t)(ρ(s + t) − ρ(s))f (s + t) dt.
(34)
R
Since |φ(t)(ρ(s+t)−ρ(s))| ≤ CkρkC α min(|t|α−1−k , |t|−1−k ), and the right hand
side is an integrable function of t, the estimate (33) follows from Lemma 1. Lemma 7. Let 0 < k < α ≤ 1 and let K ⊂ R2 be a compact set. Then there
exists a constant C such that
kJk Rρθ f − Rρθ (Jkθ f )k ≤ CkρkCθα kf k
(35)
for any weight function ρ(θ, x) of class Cθα and any f ∈ L2 (R2 ) with support in
S 1 × K and K respectively.
Proof. Note first that
Jk Rρθ f − Rρθ (Jkθ f ) = Rθ Jkθ (ρθ f ) − ρθ Jkθ f
104
III
Radon transforms and tomography
and let f˜ = Jkθ (ρθ f ) − ρθ Jk f . Write Rθ f˜ = g1 + g2 where
Z
˜ + tθ⊥ ) dt
f(sθ
g1 (s) =
|t|≤A
Z
˜ + tθ⊥ ) dt
f(sθ
g2 (s) =
|t|>A
˜ ≤
for some constant A > sup{|x|; x ∈ K}. It follows from Lemma 6√that kfk
˜
α
Ckρθ kCθ kf k, and Cauchy-Schwarz inequality implies that kg1 k ≤ 2Akfk. To
estimate the norm of g2 , note that
Z
1/2
˜ + tθ⊥ )| ≤ C|t|−1−k sup |ρ|
|f(sθ
|f (sθ + tθ⊥ )|2 dt
R
for |t| > A, which implies that kg2 k ≤ C sup |ρ|kf k.
Lemma 8. Let a and b be real numbers, not equal to −1/2. Then there exists
a constant C such that
Z 2π
(1 + r2 cos2 φ)a (1 + r2 sin2 φ)b dφ ≤ C(1 + r2 )max(a−1/2,b−1/2,a+b) (36)
0
for all real numbers r.
Proof. The statement is obviously true for small values of r, so we may assume
that r > 1. Then the change of variables t = tan φ gives
Z
2π
(1 + r2 cos2 φ)a (1 + r2 sin2 φ)b dφ
0
=4
≤C
Z
1/r
0
Z
r2a dt +
∞
0
Z
1
1/r
r2
1+
1 + t2
a r2a+2b t2b dt +
Z
r 2 t2
1+
1 + t2
r
b
dt
1 + t2
r2a+2b t−2a−2 dt +
1
Z
∞
r
r2b t−2 dt
!
≤ C 0 (r2a−1 + r2b−1 + r2a+2b )
where C and C 0 are constants depending only on a and b. The estimate follows
easily.
Lemma 9. Let k > −1, let α > 1/2 and let K ⊂ R2 be a compact set. Then
there exists a constant C such that
kgk ≤ CkρkC α⊥ kf kk
θ
(37)
for any weight function ρ(θ, x) of class Cθα⊥ vanishing for x outside K and any
f ∈ H k (R2 ), where
θ
g(θ, s) = Rρθ (Jk+1/2
f )(s).
(38)
105
III
Radon transforms and tomography
Proof. Note that it follows from Parseval’s formula that
⊥
θ⊥
Rρθ f = Rθ (ρθ f ) = Rθ (Jlθ ρθ )(J−l
f)
for any real number l. From Cauchy-Schwarz inequality it follows also that
Z
kRθ (ρθ f )k2 ≤ kf k2 sup
|ρθ (sθ + tθ⊥ )|2 dt.
s∈R
R
Fix some l with 1/2 < l < α. Then there exists a constant C, depending on
l, α and K, such that
Z 2
θ⊥
Jl ρθ (sθ + tθ⊥ ) dt ≤ Ckρk2C α⊥ .
θ
R
Hence it follows that
2
⊥
⊥
θ θ
2
θ⊥ θ
θ
Jk+1/2 f Jk+1/2 f ) ≤ Ckρk2C α⊥ J−l
kRρθ (Jk+1/2
f )k2 = Rθ (Jlθ ρθ )(J−l
θ
and that
kgk2 ≤ Ckρk2C α⊥
θ
Z
S1
2
⊥
θ θ
J−l Jk+1/2 f dθ.
We shall now estimate the integral on the right hand side of this inequality.
Writing ξ · θ = r cos φ, ξ · θ ⊥ = r sin φ and using Lemma 8 with a = k + 1/2 and
b = −l, we have
Z Z
Z 2
(1 + (ξ · θ)2 )k+1/2 ˆ 2
1
θ⊥ θ
|f(ξ)| dξ dθ
J−l Jk+1/2 f dθ =
2
(2π) S 1 R2 (1 + (ξ · θ⊥ )2 )l
S1
Z
C
≤
(1 + |ξ|2 )k |fˆ(ξ)|2 dξ
(2π)2 R2
= Ckf k2k .
This completes the proof.
Lemma 10. If k is any real number, then
√
kI1/2 Rf kk = 2 πkf kk
(39)
for any function f ∈ H k (R2 ) and
√
kR∗ I1/2 gkk = 2 πkgk
for any function g ∈ H k (S 1 × R).
106
(40)
III
Radon transforms and tomography
Proof. From (18) in Lemma 4 it follows that
Z
1
kI1/2 Rf k2k =
|σ|(1 + σ 2 )k |(Rf ˆ) (θ, σ)|2 dθ dσ
2π S 1 ×R
Z
1
ˆ
=
(1 + |ξ|2 )k |f(ξ)|
dξ
π R2
= 4πkf k2k .
This proves (39). The other equality follows similarly from (20).
Proof of Theorem 1 and Theorem 2. First we prove the estimate (26)
θ
in Theorem 1. Let g(θ, s) = Rρθ (Jk+1/2
f ). Then it follows from Lemma 7 and
Lemma 9 that
kRρ f kk+1/2 = kJk+1/2 Rρ f k
≤ kJk+1/2 Rρ f − gk + kgk
≤ C1 kρkCθα kf k + C2 kρkC α⊥ kf kk
θ
≤ CkρkC α kf kk .
This proves (26). To prove (25) we may assume, in view of Lemma 10, that
ρ(θ, x) vanishes for x outside K, and then we may as well assume that supp f ⊂
K. Then estimate follows from (26), since kI1/2 Rρ f kk ≤ kRρ f kk+1/2 . Theorem
2 is an immediate consequence of Lemma 9 with k = −1/2.
5
Stability of the inverse problem
In this section we show that f can be estimated in terms of Rρ f under certain
conditions on the weight function ρ.
5.1
Main results
Let W denote the set of all weight functions ρ satisfying the following conditions:
1. ρ(θ, x) is constant for all x outside some compact set.
2. For every θ ∈ S 1 , the function x 7→ ρ(θ, x) is of class C ∞ .
3. There exists a finite set E ⊂ S 1 such that ρ(θ, x), as well as all its partial
derivatives with respect to the x variable, are uniformly continuous on
(S 1 r E) × R2 .
For any real number k, let W k denote the set of weight functions ρ such
that there exists a sequence ρj ∈ W with sup |ρj − ρ| → 0 and kR∗ I1 Rρj −
R∗ I1 Rρ kk → 0 when j → ∞. Hence all weight functions in W are also in W k .
107
III
Radon transforms and tomography
In Section 5.2 we give some other sufficient conditions for a weight function to
be in W k for some values of k. Note that if ρ ∈ W k and θ0 ∈ S 1 , the the
one-sided limits
ρ(θ0± , x) = lim ρ(θ, x)
θ→θ0±
always exist. Here we take the counter-clockwise direction along the unit circle
as the positive direction. For the sake of brevity, we will refer to ρ(θ − , x) and
ρ(θ+ , x) as the left and right limits of ρ at (θ, x).
For a weight function ρ ∈ W k , let Dρ (θ, x) denote the closed disc in the
complex plane, with the segment from ρ(θ + , x) to ρ(θ− , x) as one of its diameters. If ρ is continuous at (θ, x), Dρ (θ, x) is just a point. Also let Dρ denote
the closure of the union of all Dρ (θ, x).
1 ∗
R I1 Rρ is a
4π
k
2
bounded operator on H (R ), and its essential spectrum is contained in Dρ .
Theorem 3. Let ρ ∈ W k be a real even weight function. Then
The proof is rather lengthy, and we give it in Section 5.3. From this theorem
it is possible to obtain stability results for the inverse problem of the weighted
Radon transform. We give two examples.
Theorem 4. Let ρ ∈ W k be a real even weight function, bounded from below by
a positive constant. If Rρ is injective on H k (R2 ), then there exists a constant
C such that
kf kk ≤ CkRρ f kk+1/2
(41)
for every function f ∈ H k (R2 ).
Proof. Since ρ is bounded from below by a positive constant, Dρ does not
contain the origin, so R∗ I1 Rρ is invertible in the Calkin algebra by Theorem 3.
Since Rρ is injective, and R∗ I1 is injective on the set of all even functions by
(20), and the range of Rρ consists of even functions, it follows that R∗ I1 Rρ is
injective. By Atkinson’s theorem (see for example [4, Theorem 14.2]), R ∗ I1 Rρ
has a bounded left inverse. Since R∗ I1 is a bounded operator from H k+1/2 (S 1 ×
R) to H k (R2 ) by Lemma 10, the desired estimate follows.
For practical applications the following result is of particular interest. Let
1
1
S+
denote one half of the unit circle, for example the right half: S+
= {θ =
(θ1 , θ2 ); θ1 ≥ 0}.
Theorem 5. Let µ be an attenuation function which is Hölder continuous with
positive exponent and let K ⊂ R2 be a compact set. Then there exists a constant
C depending only on µ and K such that
kf k−1/2 ≤ CkRµ f |S+1 ×R k
for every function f with support in K.
108
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Radon transforms and tomography
Proof. Let ρ0 be the weight function corresponding to µ. By Corollary
3 below there exists a positive even weight function ρ ∈ W −1/2 such that
√
1
ρ(θ, x) = ρ0 (θ, x) for all (θ, x) ∈ S+
× K, and then kRρ f k = 2kRµ f |S+1 ×R k
when supp f ⊂ K. Moreover, by a result of Novikov (see [10] or [2, Theorem 3])
it is known that Rµ is injective on the set of compactly supported functions f
(this is where the Hölder continuity of µ is needed). The estimate then follows
from Theorem 3 and Atkinson’s theorem as in the previous proof.
Corollary 2. Let µ be a Hölder continuous attenuation function, let f ∈
H −1/2 (R2 ), and assume that supp µ and supp f are both contained in a compact
set K. Let µj be a sequence of attenuation functions and fj ∈ H −1/2 (R2 ) a
sequence of functions, all with support in K, such that kµj − µkL∞ → 0 and
k(Rµj fj −Rµ f )|S+1 ×R k → 0 when j → ∞. Then kfj −f k−1/2 → 0 when j → ∞.
In fact, there exist positive constants C and δ, depending only on µ and K such
that
kfj − f k−1/2 ≤ C kµj − µkL∞ kf k−1/2 + k(Rµj fj − Rµ f )|S+1 ×R k
(43)
whenever kµj − µkL∞ ≤ δ.
Proof. By Theorem 5 there exists a constant C1 such that kfj − f k−1/2 ≤
C1 kRµ (fj − f )|S+1 ×R k, and by Corollary 1 there exists a constant C2 such that
kRµj fj − Rµ fj k ≤ C2 kµj − µkL∞ kfj k−1/2 . Hence it follows that
kfj − f k−1/2 ≤ C1 kRµ (fj − f )|S+1 ×R k
≤ C1 (kRµ fj − Rµj fj k + k(Rµj fj − Rµ f )|S+1 ×R k)
≤ C1 C2 kµj − µkL∞ (kfj − f k−1/2 + kf k−1/2)
+ k(Rµj fj − Rµ f )|S+1 ×R k .
Now (43) follows if we take δ so small that C1 C2 δ < 1/2 and C larger than both
2C1 and 2C1 C2 .
5.2
Some sufficient conditions for a weight function to be
in W k
Here we give some sufficient conditions for a weight function ρ to be in the class
W k defined in the previous section.
Theorem 6. Let 0 ≤ k < 1/2 and let k + 1/2 < α ≤ 1. If ρ(θ, x) is a weight
function, constant for x outside a compact set, piecewise continuous with respect
to θ, and kρkC α < ∞, then ρ ∈ W k .
109
III
Radon transforms and tomography
Proof. It follows from Lemma 10 and Theorem 1 that if 0 ≤ k < 1/2, k +
1/2 < α ≤ 1, and ρ(θ, x) is constant for x outside some compact set K, then
kR∗ I1 Rρ kk ≤ CkρkC α , where the constant C depends only on k, α and K.
Suppose now that ρ satisfies the hypothesis of Theorem 6, and take a number
α0 with k + 1/2 < α0 < α. Then there exists a sequence of weight functions
ρj ∈ W such that kρj − ρkC α0 → 0, and hence kR∗ I1 Rρj − R∗ I1 Rρ kk → 0. This
shows that ρ ∈ W k , and proves the theorem.
Theorem 7. Let α > 1/2 and let ρ(θ, x) be a weight function, constant for x
outside a compact set, continuous with respect to x, piecewise continuous with
respect to θ and such that kρkC α⊥ < ∞. Then ρ ∈ W −1/2 .
θ
Proof. This follows from Theorem 2 in the same way as the previous proof.
Corollary 3. Let µ be a continuous attenuation function, let ρ0 be the corresponding weight function and let K ⊂ R2 be a compact set. Then there exists
a positive even weight function ρ ∈ W −1/2 such that ρ(θ, x) = ρ0 (θ, x) for all
1
(θ, x) ∈ S+
× K.
5.3
Proof of Theorem 3
It is sufficient to prove the theorem for weight functions in W, since both Dρ
and the essential spectrum converge when passing to the limits used to define
Wk.
Lemma 11. Let ρ ∈ W be an even weight function and assume that 0 ∈
/
Dρ . Then there exists a positive constant a < 1 and a continuous even weight
function ρ̃ ∈ W such that
1
¯ x) − 2| + sup |(ρ(θ+ , x) − ρ(θ− , x))ρ̃(θ,
¯ x)| ≤ a
sup |(ρ(θ+ , x) + ρ(θ− , x))ρ̃(θ,
2 x
x
(44)
for all θ ∈ S 1 . In fact, this is possible with
a=1−
dist(0, Dρ )
sup{|z|; z ∈ Dρ }
(45)
where dist(0, Dρ ) denotes the distance from the origin to Dρ .
Note that at points θ where ρ is continuous, the condition on ρ̃ is that
¯ x) − 1| ≤ a.
|ρ(θ, x)ρ̃(θ,
110
(46)
III
Radon transforms and tomography
Proof. Note that the a defined by (45) satisfies
a ≥ sup
x,θ
|ρ(θ+ , x) − ρ(θ− , x)|
.
|ρ(θ+ , x) + ρ(θ− , x)|
1
At all θ ∈ S where ρ has a jump, we take
¯ x) =
ρ̃(θ,
ρ(θ+ , x)
2
.
+ ρ(θ− , x)
Then (44) is clearly satisfied at these points. At other points the optimal choice
¯ x) = 1/ρ(θ, x), but this would violate the continuity of ρ̃.
would be to take ρ̃(θ,
Instead we modify the definition near each discontinuity θ0 as follows. Take a
cutoff function ϕ with ϕ(θ0 ) = 1 and vanishing outside a small neighborhood of
θ0 . For θ slightly to the right of θ0 , define
¯ x) =
ρ̃(θ,
2
(2 − ϕ(θ))ρ(θ, x) + ϕ(θ)ρ(θ0− , x)
and for θ to the left of θ0 we use an analogous construction. Provided that ϕ
has sufficiently small support, the condition (44) will then be satisfied at all
points.
For the rest of the lemmas in this section we will use the following notation.
Assume that ρ is a weight function in W and that 0 ∈
/ Dρ and let ρ̃ be a weight
function as in Lemma 11. Define
¯ 0 , x) − 1,
β(θ, θ0 , x) = ρ(θ, x)ρ̃(θ
Pρ =
and
T = Pρ Pρ̃∗ − Id =
as in Lemma 5, and let
(47)
1 ∗
R I1 R ρ
4π
(48)
1
R∗ I1 Rρ Rρ̃∗ I1 R − Id
16π 2
(49)
r0 (ξ) = sup β̂(θ, θ0 , ξ) .
(50)
θ,θ 0
Let A and B be two positive numbers. According to Lemma 5, T can be written
as the sum of the three operators Tj defined by
Z
ξ η
1
ˆ
(51)
β̂
, , ξ − η fˆ(η) dη
(T1 f ) (ξ) =
(2π)2 |ξ−η|>A
|ξ| |η|
Z
1
ξ η
(T2 f ˆ) (ξ) =
fˆ(η) dη
(52)
β̂
,
,
ξ
−
η
(2π)2 |ξ−η|≤A
|ξ|
|η|
|η|>B
Z
1
ξ η
ˆ
ˆ dη.
(T3 f ) (ξ) =
(53)
β̂
, , ξ − η f(η)
(2π)2 |ξ−η|≤A
|ξ|
|η|
|η|≤B
111
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Radon transforms and tomography
Lemma 12. If T1 is defined by (47) and (51) where ρ and ρ̃ are in W, then
kT1 kk → 0 when A → ∞.
Proof. Let
r(ζ) = sup |β̂(θ, θ0 , ζ)|
θ,θ 0 ,ξ
∞
(1 + |ξ|2 )k/2
.
(1 + |ξ − ζ|2 )k/2
Since β is of class C with respect to the x variable, it follows that r(ζ) decreases
faster than any power of |ζ| when |ζ| → ∞. In particular, r is integrable and it
follows from Lemma 1, applied to the Fourier transforms, that
Z
r(ζ) dζ → 0
kT1 kk ≤
|ζ|>A
when A → ∞.
Lemma 13. If T2 is defined by (47) and (52) where ρ and ρ̃ are in W, ρ̃ is
continuous and (44) holds, then
Z
2
lim sup kT2 kk ≤ a +
r0 (ξ) dξ
(2π)2 |ξ|>A
B→∞
for any fixed A > 0.
Proof. Note that by Lemma 3, it is sufficient to consider the case k = 0.
Let δ be an arbitrary positive number, and take a positive number so small
that
Z
(54)
sup β̂(θ, θ, ξ) − β̂(θ0 , θ00 , ξ) dξ < δ
|ξ|≤A θ,θ 0 ,θ 00
where the supremum is taken over all θ, θ 0 and θ00 which lie on an arc of length
smaller than , and moreover ρ is continuous on the subarc spanned by θ and
θ0 .
Let N be a positive integer with 1/N < δ, and let χj be functions on S 1
such that
0 ≤ χj (θ) ≤ 1
and
X
χj (θ) = N.
j
We let χ̃j denote a nonnegative function, which is equal to 1 on a neighborhood
of supp χj . The functions χj and χ̃j can be chosen to satisfy the following
conditions. First
X
χ̃j (θ) ≤ N + 1
j
112
III
Radon transforms and tomography
and moreover, supp χ̃j is an arc of length at most and χ̃j is equal to 1 on an
/(4N )-neighborhood of supp χj . To see this, partition S 1 into arcs of length
between /(2N ) and /(N + 1), and let each χj be equal to 1 on N consecutive
arcs and 0 elsewhere. Let χ̃j be equal to 1 on supp χj and on one half each of
the two neighboring arcs, and 0 at all other points.
We may assume that each supp χ̃j contains at most one discontinuity of ρ,
otherwise we just take a smaller . Pick a point θj ∈ supp χ̃j for each j, which
−
we assume to be at the discontinuity of ρ if there is one. Let χ+
j and χj be
equal to χj to the right and left of θj respectively and 0 on the other side.
Choose B so large that the distance between ξ/|ξ| and η/|η| is at most
/(4N ) whenever |ξ − η| ≤ A and |η| > B. Define operators Sj by
Z
ξ
1
χ
(Sj f ˆ) (ξ) =
β̂A (θj , θj , ξ − η)fˆB (η) dη
(55)
j
(2π)2
|ξ|
if ρ is continuous at θj , and
(Sj f ˆ) (ξ) =
1
χ+
j
(2π)2
+
ξ
|ξ|
χ−
j
Z
ξ
|ξ|
β̂A (θj+ , θj , ξ − η)fˆB (η) dη
Z
β̂A (θj− , θj , ξ
− η)fˆB (η) dη
!
(56)
otherwise. Here fˆB (η) = fˆ(η) if |η| > B and 0 otherwise, and β̂A (θ, θ0 , ξ) =
R
β̂(θ, θ0 , ξ) if |ξ| ≤ A and 0 otherwise. Since |β − βA | ≤ |ξ|>A r0 (ξ) dξ/(2π)2 , the
condition (46) implies that
Z
2
r(ξ) dξ
sup |βA (θj , θj , ξ)| ≤ a0 := a +
(2π)2 |ξ|>A
ξ
if ρ is continuous at θj . Since the operator Sj consists of multiplication by
βA (θj , θj , ·) followed by multiplication on the Fourier transform with χj (ξ/|ξ|),
it follows that
kSj k ≤ a0 .
When ρ is discontinuous at θj , it follows from (44) that
1
sup |βA (θj+ , θj , ξ) + βA (θj− , θj , ξ)| + sup |βA (θj+ , θj , ξ) − βA (θj− , θj , ξ)| ≤ a0
2 ξ
ξ
and an application
of Lemma 2 gives again the estimate kSj k ≤ a0 .
1 P
Let S = N j Sj . Then it follows from (54) and Lemma 1 that kT2 −
Sk ≤ δ/(2π)2 . Consider arbitrary L2 -functions f and f˜. Define fj by fˆj (ξ) =
P
χ̃j (ξ/|ξ|)fˆ(ξ) and define f˜j similarly. Since j |fj | ≤ (N + 1)|f | and hfj , fj i =
113
III
Radon transforms and tomography
P
P
2
2
hfj , f i it follows that
j hfj , f i ≤ (N + 1)kf k , and similarly,
j kfj k =
P ˜ 2
2
˜
j kfj k ≤ (N + 1)kfk . Then
D
E
E
1 X D
Sf, f˜ ≤
Sj f, f˜ N j
E
1 X D
=
Sj fj , f˜j N j
a0 X
kfj kkf˜j k
N j
s
X
a0 X
kf˜j k2
≤
kfj k2
N
j
j
≤
a0 (N + 1)
˜
kf kkfk
N
˜
≤ a0 (1 + δ)kf kkfk
≤
and it follows that kSk ≤ a0 (1 + δ). We conclude that kT2 k ≤ a0 (1 + δ) + δ/(2π)2
for sufficiently large B, and since δ was arbitrary, the result follows.
Corollary 4. If ρ and ρ̃ are weight functions in W, ρ̃ is continuous, and (44)
holds, then kPρ Pρ̃∗ − Id kCk ≤ a.
Proof. Consider the decomposition T = T1 + T2 + T3 defined by (51), (52)
and (53). Here T3 is a Hilbert-Schmidt operator, and therefore compact. It
follows that kT kCk ≤ kT1 kk + kT2 kk and by Lemmas 12 and 13 this can be
made arbitrarily close to a by taking A and B sufficiently large.
Theorem 3 is a direct consequence of the following result.
Corollary 5. If ρ is a weight function in W and λ ∈ CrDρ , then Pρ −λ Id has a
right inverse in the Calkin algebra whose norm is not greater than sup{|z−λ|; z ∈
Dρ }/ dist(λ, Dρ )2 . If λ is in the unbounded component of C r Dρ , this right
inverse is also a left inverse.
Proof. Since Pρ − λ Id = Pρ−λ , it is sufficient, for the first statement, to
consider λ = 0. So assume that 0 ∈
/ Dρ and take a and ρ̃ as in Lemma 11.
By Corollary 4, Pρ Pρ̃∗ has an inverse in the Calkin algebra whose norm is not
greater than 1/(1 − a) = sup{|z|; z ∈ Dρ }/ dist(0, Dρ ). Applying Corollary 4
with ρ ≡ 1, it follows that kPρ̃ kCk ≤ sup |ρ̃| ≤ 1/ dist(0, Dρ ), and this proves the
first claim.
To prove the second claim, note that our right inverse must be the inverse
of Pρ − λ Id for λ outside the essential spectrum of Pρ , in particular for all
sufficiently large λ. Since the norm of (Pρ − λ Id)−1 tends to infinity when λ
114
III
Radon transforms and tomography
approaches the essential spectrum, and the norm of our right inverse is locally
bounded on C r Dρ , it follows that each connected component of C r Dρ is
either contained in the essential spectrum or its complement. This completes
the proof.
6
Inversion formula for the exponential Radon
transform
Here we derive a new inversion formula for the exponential Radon transform.
This formula differs from the well known formula of Tretiak and Metz in that
it requires the Radon transform Rµ f (θ, s) to be known only for θ on a half of
the unit circle. In this section, µ denotes a real number and Rµ denotes the
corresponding exponential Radon transform.
6.1
Main results
Our inversion formula is derived as a corollary of three theorems. The proofs of
these are given in the next section.
Theorem 8. If f ∈ C01 (R2 ), then
Z
Z ∞
⊥
∂Rµ f
cosh(µt)
(θ, x · θ)e−µx·θ dθ = 2
dt
f (x1 + t, x2 )
1
∂s
t
S+
−∞
(57)
where the singularity at t = 0 in the integral on the right hand side is treated as
a principal value.
Let chµ denote the distribution
chµ (t) =
cosh(µt)
t
with a principal value at the origin. (Later, we will also use chµ to denote the
meromorphic function defined by the same expression.) The next step is to find
a compactly supported distribution u such that the convolution u∗chµ restricted
to a given compact set is a point mass δ0 at the origin. This is transformed into
a problem about functions of one complex variable by means of the following
definitions.
Let ϕ be a function, holomorphic in the whole complex plane except on some
subset of the real line. Define B+ ϕ and B− ϕ to be the boundary values of ϕ on
the real line from above and from below:
B± ϕ(t) = lim ϕ(t ± i)
&0
115
(58)
III
Radon transforms and tomography
provided that these limits exist as distributions, and let BΣ ϕ = B+ ϕ + B− ϕ
and B∆ ϕ = B+ ϕ − B− ϕ.
Furthermore, if ϕ and ψ are holomorphic outside some compact set in the
complex plane, define an entire function [ϕ, ψ] by the formula
Z
1
[ϕ, ψ](z) =
ϕ(ζ)ψ(z − ζ) dζ
(59)
2πi Γ
where Γ is a closed curve, depending on z, so large that the integrand is holomorphic in the unbounded component of C r Γ.
Theorem 9. Let r < R be positive numbers, and let F be a function holomorphic in C r [−R, R]. If
BΣ F (t) = −
1
δ0 (t)
2πi
for |t| < r
(60)
and
[chµ , F ] ≡ 0,
(61)
then u = B∆ F satisfies 2(u ∗ chµ )(t) = δ0 (t) for |t| < r.
Finally, we show that it is possible to find functions F satisfying the hypothesis of Theorem 9. See section 6.3 for some remarks on the computation of
F.
Theorem 10. Let w(α) be a positive function on the interval [0, 1], and let
G(z) be defined for z ∈ C r [−R, R] by
G(z) =
Z
1
0
p
w(α)
dα,
r2 + (R2 − r2 )α − z 2
(62)
√
where for any positive real number c, we have
the branch of√1/ c − z 2
√ chosen
√
which is holomorphic outside the interval [− c, c] and satisfies z/ c − z 2 → i
when z → ∞. For generic choice of r and R there exist a number a and an odd,
entire function h such that
a
F (z) =
+ h(z) G(z)
(63)
z
satisfies the hypothesis (60) and (61) in Theorem 9.
Remark. It is likely that the conclusion of the theorem holds for all r, R and
w, but we do not have a proof.
116
III
Radon transforms and tomography
Combining these results, we obtain an inversion formula for the exponential
Radon transform.
Corollary 6. Let D ⊂ R2 be a compact set, and let r ≥ sup{|x1 −y1 |; x, y ∈ D}.
Choose a number R > r and a positive function w on the interval [0, 1], let F be
the function constructed in Theorem 10 which satisfies the hypothesis of Theorem
9, and let u = B∆ F . If f is any function of class C 2 with supp f ⊂ D, then
f (x) =
Z
R
u(t)
−R
Z
1
S+
∂Rµ f
θ, θ1 (x1 + t) + θ2 x2 e−µ(θ1 (x1 +t)+θ2 x2 ) dθ dt (64)
∂s
for all x ∈ D.
6.2
Proofs.
Proof of Theorem 8. Let ∂θ denote the directional derivative in direction θ.
Then it follows from the definition of the exponential Radon transform, that
Z ∞
∂Rµ f
(θ, s) =
∂θ f (sθ + tθ⊥ )eµt dt
∂s
−∞
and hence
Z Z ∞
Z
⊥
∂Rµ f
−µx·θ ⊥
(θ, x · θ)e
dθ =
∂θ f ((x · θ)θ + tθ⊥ )eµt−µx·θ dt dθ
1
1
∂s
S+ −∞
S+
Z Z ∞
∂θ f (x + τ θ⊥ )eµτ dτ dθ
=
=
Z
1
−∞
S+
∞ Z
−∞
1
S+
∂θ f (x + τ θ⊥ )eµτ dθ dτ
where we have made the change of variables τ = t − x · θ ⊥ . For fixed x and τ ,
1
θ is the tangent vector to the semicircle {x + τ θ ⊥ ; θ ∈ S+
}, so that
Z
1
∂θ f (x + τ θ⊥ ) dθ =
f (x1 + τ, x2 ) − f (x1 − τ, x2 ) .
1
τ
S+
Combining these computations, it follows that
Z ∞
Z
∂Rµ f
1
−µx·θ ⊥
(θ, x · θ)e
dθ =
(f (x1 + τ, x2 ) − f (x1 − τ, x2 ) eµτ dτ
1
∂s
−∞ τ
S+
Z
eµτ + e−µτ
= lim
f (x1 + τ, x2 ) dτ
→0 |τ |>
τ
117
III
Radon transforms and tomography
Proof of Theorem 9. The theorem is a direct consequence of the following
identity.
Lemma 14. If ϕ and ψ are holomorphic outside a compact subset of the real
line, then
1
BΣ ϕ ∗ B ∆ ψ − B ∆ ϕ ∗ B Σ ψ .
(65)
[ϕ, ψ] =
4πi
R
Proof. Let z ∈ R and suppose that ϕ(ζ) and ψ(z − ζ) are holomorphic for ζ
outside the interval [a, b]. Then
1
[ϕ, ψ](z) = lim
→0 2πi
Z
b
a
+
=
1
2πi
Z
Z
b
a
ϕ(t − i)ψ(z − (t − i)) dt
Z
a
b
ϕ(t + i)ψ(z − (t + i)) dt
!
B− ϕ(t)B+ ψ(z − t) − B+ ϕ(t)B− ψ(z − t) dt
b
1
BΣ ϕ(t)B∆ ψ(z − t) − B∆ ϕ(t)BΣ ψ(z − t) dt
4πi a
1
=
(BΣ ϕ ∗ B∆ ψ)(z) − (B∆ ϕ ∗ BΣ ψ)(z) .
4πi
=
To prove the theorem, take ϕ = chµ and ψ = F . Then BΣ ϕ = 2 chµ and
B∆ ϕ = −2πiδ0 . From the assumption that [chµ , F ] = 0 it follows that
2 chµ ∗u = BΣ chµ ∗B∆ F = B∆ chµ ∗BΣ F = −2πiδ0 ∗ BΣ F.
The conclusion of Theorem 9 follows from the assumption on BΣ F .
Proof of Theorem 10. We must show that there exist a number a and an
entire function h such that
a
+ h(z) G(z)
F (z) =
z
satisfies (60) and (61). Note that for ϕ, ψ holomorphic outside the real line,
BΣ (ϕψ) =
1
BΣ ϕBΣ ψ + B∆ ϕB∆ ψ
2
118
(66)
III
Radon transforms and tomography
provided that the products on the right make sense. Since BΣ G = B∆ h = 0 in
(−r, r), it follows that in the interval (−r, r),
a
1
BΣ F = B∆
B∆ G = −πiaB∆ G(0)δ0
2
z
!
Z 1
w(α)
p
dα δ0 .
= −2πia
r2 + (R2 − r2 )α
0
Hence, the condition (60) will be satisfied precisely if
a=−
4π 2
R1
0
1
√
w(α)
r 2 +(R2 −r 2 )α
dα
.
(67)
It remains to determine h so that (61) is satisfied. To prove the existence of h
it is useful to reformulate the condition by means of the following lemma.
Lemma 15. Let h and ψ be entire holomorphic functions, and let ϕ be holomorphic
−1
outside a compact set, with a zero of
order k ≥ 0 at infinity. Then
z , hϕ = ψ if and only if h = z −1 , ψ/ϕ + P for some polynomial P of
degree at most k − 1.
Proof. Note first that for any ϕ holomorphic outsidea compact
set, z −1 , ϕ is
the unique entire function with the property that ϕ − z −1 , ϕ = O(|z|−1 ) when
z → ∞. From this it follows that
z −1 , hϕ = ψ ⇐⇒ hϕ − ψ = O(|z|−1 ) ⇐⇒
h − ψ/ϕ = O(|z|k−1 ) ⇐⇒ h − z −1 , ψ/ϕ is a polynomial of degree at most
k − 1.
Rewrite the condition (61) as
1
cosh(µz) − 1
aG(z)
.
, hG = −
, hG − chµ ,
z
z
z
(68)
Since G has a zero of order 1 at ∞, it follows from Lemma 15 that this condition
will be satisfied if
aG(z)
1 1 cosh(µz) − 1
1 1
h=− ,
chµ ,
.
(69)
, hG − ,
z G
z
z G
z
Write the right hand side of (69) as −Φ(h) − H where H is an entire function
and Φ is a linear operator on the space of entire functions. Let K be a compact
set containing the interval [−R, R] in its interior, and let A(K) be the Banach
space of functions continuous in K and holomorphic in the interior of K. Since
(cosh(µz)−1)/z is an entire function, the contour of integration in the definition
of
Z
cosh(µz) − 1
1
cosh(µζ) − 1
, hG =
h(z − ζ)G(z − ζ) dζ
z
2πi Γ
ζ
119
III
Radon transforms and tomography
can be chosen so that the argument of h always is in the interior of K. From
this it is clear that Φ can be extended to a bounded operator from A(K) to
A(K 0 ) for any compact set K 0 ⊂ C. If K is contained in the interior of K 0 ,
the restriction A(K 0 ) → A(K) is compact, and it follows that Φ is a compact
operator on A(K). So unless −1 happens to be an eigenvalue of Φ, the equation
h + Φ(h) = −H has a unique solution h ∈ A(K). Since both H and Φ(h) are
entire functions, it follows that h is also entire. Since Φ takes odd functions to
odd functions, even functions to even functions and H is odd, it follows that h
is odd.
Finally, note that Φ depends analytically on r and R, and the norm of Φ
converges to 0 when r and R approach 0. Hence, −1 is not an eigenvalue of
Φ for generic choices of r, R and w. In fact, numerical experiments seem to
suggest that the eigenvalues of Φ are always positive.
6.3
Numerical test.
To use the inversion formula on numerical data, it is necessary to choose a weight
w, compute approximations for the corresponding a and h to find a function
F = (a/z + h)G satisfying the hypothesis of Theorem 9, and then compute a
list of values of u = B∆ F .
Choice of w. Choosing w to be a piecewise linear function makes the computation of G straightforward. In order to make G fairly smooth, it is advisable
to make w(0) = w(1) = 0.
Computation of a and h. The constant a is found directly from (67). The
function h is computed directly from the equation (61) rather than (69). More
precisely, h is approximated by an odd polynomial. Choose a positive integer
e µ and G
e be the Laurent series expansions of chµ and G up to terms
n, and let ch
of some finite degree, say 4n. Use the relations

k

j+1

(−1)
z j+k+1 if j < 0 and j + k + 1 ≥ 0


−1 − j


j k
j
z ,z =
k
z j+k+1
if k < 0 and j + k + 1 ≥ 0
 (−1)


−1 − k



0
otherwise
(70)
and the bilinearity of [· , ·] to compute an odd degree 2n + 1 polynomial hn such
that the Taylor series of
i
h
e µ , a + hn G
e
ch
z
vanishes up to terms of degree 2n. Note that this expression is an even function,
so we have n + 1 linear equations in the n + 1 unknown coefficients of hn . Then
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III
Radon transforms and tomography
it is easy to show that if a solution h of (69) exists, hn converges to h when
n → ∞.
Computation of u. The distribution u is readily computed by the formula
a
Z 1
w(α)
p
u(t) = 2
+ hn (t)
dα (71)
2
t
r + (R2 − r2 )α − t2
max(0,(t2 −r 2 )/(R2 −r 2 ))
Here u is treated as a function rather than a distribution. When computing
numerically the integral in (64), it is necessary to deal with the singularity of
u at the origin. One simple-minded approach is to use the trapezoid rule on a
set of nodes symmetric with respect to the origin to approximate the principal
value integral. More accurate results can be obtained by using the methods
described in [7, Chapter III].
The following reconstruction was made with the values r = 1, R = 1.5 and 10
nonzero terms in the polynomial hn . The test object consists of circular discs,
and the Radon transform was sampled at 200 values of θ equally spaced over
1
S+
, and 101 values of s equally spaced between −0.5 and 0.5. The width and
height of the image are 1 and the attenuation µ = 3.
References
[1] J. Boman: An example of nonuniqueness for a generalized Radon transform, J. Anal. Math., 61 (1993), 395–401.
[2] J. Boman, J. O. Strömberg: Novikov’s inversion formula for the attenuated
Radon transform — a new approach. To appear.
[3] V. Dicken: A new approach towards simultaneous activity and attenuation
reconstruction in emission tomography, Inverse Problems 15 (1999), no. 4,
931–960.
[4] P. R. Halmos, V. S. Sunder: Bounded integral operators on L2 spaces,
Springer-Verlag, 1978.
[5] P. Kuchment, I. Shneiberg: Some inversion formulae in the single photon
emission computed tomography, Appl. Anal. 53 (1994), 221–231.
[6] A. Markoe, E. T. Quinto: An elementary proof of local invertibility for
generalized and attenuated Radon transforms, SIAM J. Math. Anal. 16
(1985), no. 5, 1114–1119.
[7] F. Natterer: The mathematics of computerized tomography, Wiley, 1986.
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Original image
Reconstruction
0.5
0.5
0
0
−0.5
−0.5
0
0.5
−0.5
−0.5
0
0.5
Figure 1: Exact image and reconstruction obtained using the inversion formula.
Cross section along x −axis
Cross section along x −axis
1
2
2
2
1.5
1.5
1
1
0.5
0.5
0
−0.5
0
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
Figure 2: Cross section of exact image (dotted) and reconstruction (solid) along
the horizontal (left) and vertical (right) lines through the center of the image.
[8] F. Natterer: Inversion of the attenuated Radon transform, Inverse Problems 17 (2001), 113–119.
[9] F. Noo, J. M. Wagner: Image reconstruction in 2D SPECT with 180◦
acquisition, Inverse Problems 17 (2001), 1357–1371.
[10] R. G. Novikov: An inversion formula for the attenuated X-ray transformation, Ark. Mat. 40 (2002), 145–167.
[11] J. Radon: Über die Bestimmung von Funktionen durch ihre Integralwerte
längs gewisser Mannigfaltigkeiten, Ber. Ver. Sächs. Akad. 69 (1917), 262–
277
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[12] H. Rullgård:
An explicit inversion formula for the exponential Radon transform using data from 180◦ ,
Preprint:
http://www.math.su.se/reports/2002/9. To appear in Ark. Mat.
[13] O. Tretiak, C. Metz: The exponential Radon transform, SIAM J. Appl.
Math. 39 (1980), 341–354.
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