Download Introduction to amoebas and tropical geometry

Introduction to amoebas and tropical
geometry
Milo Bogaard
17-2-2015
Master Thesis
Supervisor: dr. Raf Bocklandt
KdV Instituut voor wiskunde
Faculteit der Natuurwetenschappen, Wiskunde en Informatica
Universiteit van Amsterdam
Abstract
In this thesis we give an introduction to the theory of tropical
geometry and its applications to amoebas. We treat Kapranov’s
theorem and Mikhalkin’s limit construction for amoebas.
We also compute a number of examples.
Information
Title: Introduction to amoebas and tropical geometry
Author: Milo Bogaard, [email protected], 5743117
Supervisor: dr. Raf Bocklandt
Second reviewer: Hessel Posthuma
Submitted: 17-2-2015
Korteweg de Vries Instituut voor Wiskunde
Universiteit van Amsterdam
Science Park 904, 1098 XH Amsterdam
http://www.science.uva.nl/math
Contents
Introduction
3
1 Polyhedral complexes and tropical curves
1.1 Polyhedral geometry . . . . . . . . . . . .
1.2 The Newton polytope and its subdivisions
1.3 Tropical algebra . . . . . . . . . . . . . . .
1.4 Tropical hypersurfaces . . . . . . . . . . .
1.5 The Newton polytope of a tropical curve .
.
.
.
.
.
6
. 6
. 8
. 11
. 13
. 15
.
.
.
.
.
.
.
.
19
19
21
23
26
3 Puiseux series and Kapranov’s theorem
3.1 Valuations and Puiseux series . . . . . . . . . . . . . . . . . .
3.2 Tropical polynomials . . . . . . . . . . . . . . . . . . . . . . .
3.3 Kapranov’s theorem . . . . . . . . . . . . . . . . . . . . . . .
27
27
29
32
4 A limit of amoebas
4.1 The construction of the limit . . . . .
4.2 The Hausdorff distance . . . . . . . .
4.3 A neighborhood of the tropical curve
4.4 The limit . . . . . . . . . . . . . . .
36
36
38
38
40
2 The
2.1
2.2
2.3
2.4
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
amoeba of a planar curve
Definitions and examples . . . . . . . . . . . . . . .
Laurent series . . . . . . . . . . . . . . . . . . . . .
The amoeba and the Newton polytope . . . . . . .
The components of the complement of the amoeba
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
5 Appendix I
44
5.1 Endomorphisms of (C∗ )2 . . . . . . . . . . . . . . . . . . . . . 44
5.2 The transformation of the amoeba . . . . . . . . . . . . . . . . 47
1
6 Appendix II
49
6.1 Computing the amoeba of a planar curve . . . . . . . . . . . . 49
6.2 Critical points . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
Populaire samenvatting(Dutch)
54
2
Introduction
In this thesis we give an introduction to the related subjects of amoebas and
tropical geometry.
If X is an algebraic variety in the algebraic torus (K ∗ )n over a field K
with a norm | · |K the amoeba is the image of X under the map defined
by LogK (z1 , ..., zn ) = (log(|z1 |K ), ...., log(|z1 |K )). This construction was first
used in 1971 by George Bergman to proof a theorem about subgroups of
GL(n, Z) in [2]. We briefly explain the relation between amoebas and the
group GL(n, Z) in appendix I.
The name ’amoeba’ was introduced by Gelfand, Kapranov and Zelevinsky
who rediscovered the concept when studying the combinatorics of discriminants of polynomials in [8]. The name is very appropriate considering for
example figure 1.
Figure 1: An example of an amoeba of a curve in (C∗ )2 .
3
Tropical geometry is a concept from computer science and is named after
the Brazilian mathematician and computer scientist Imre Simon. A tropical
curve in R2 is the set of points where a finite collection of linear functions
does not have a unique maximum, for example figure 2.
Figure 2: An example of a tropical curve.
It turns out that many results from algebraic geometry also hold for tropical curves, for example Bezout’s theorem for the number of intersections of
two curves. Many results of this type can be found in [21] and [16].
For fields with a non-Archimedean absolute value the relation between
amoebas and tropical curves is very direct, the amoeba of a hypersurface
is given by a tropical hypersurface. This is Kapranov’s theorem, which we
prove in chapter 3.
The relation between curves over the complex numbers and amoebas is
more complicated. Using complex analysis Mikhail Passare and Hans Rullgard constructed a tropical curve which is a deformation retract of a given
amoeba. This construction can be found in [18].
In the last chapter of this thesis we study Mikhalkin’s theorem on the
limit of a sequence of scaled amoebas from the article [14].
A nice application of this theorem is the construction of amoebas with a
prescribed number of components. For example the theorem shows that the
scaled amoeba of f = u3 + w3 + u2 w4 + u4 w3 + ru3 w + r2 u2 w2 + ruw3 + ru3 w3
converges to figure 2 for r → ∞.
4
If we pick r = 4 we already get the four bounded components we wanted.
We can change the coefficients somewhat without changing the topology
of the amoeba and figure 1 is the amoeba of the curve in (C∗ )2 given by
u3 + w3 + u2 w4 + u4 w3 + 3.8u3 w + 12.9u2 w2 + 4.55uw3 + 4.6u3 w3 .
A much more serious application of amoebas is to study the topology and
analytic structure of real and complex varieties as in [20], [14] and [15] by
Viro and Mikhalkin.
5
Chapter 1
Polyhedral complexes and
tropical curves
In this chapter we give a brief introduction to tropical geometry. The goal
is to show the connection between tropical hypersurfaces and subdivisions of
polytopes, which is theorem 1.28.
1.1
Polyhedral geometry
In this section we state a number of definitions of polyhedral geometry. Most
of this section is derived from the excellent reference work [26].
A subset X of Rn is called convex if for every pair of points x, y ∈ X the
line segment connecting x and y is contained in X. The convex hull of a set
A ⊂ Rn is the intersection of all convex sets which contain A and is denoted
by conv(A).
Definition 1.1. A set P ⊂ Rn is called a polytope if it is the convex hull a
finite set of points.
The following concrete description of the set of points of a polytope is
sometimes useful.
n
Proposition
P1.2. If A ⊂ R is a finite set then conv(A)
P consists of all points
of the form v∈B λv v with B ⊆ A, λv ∈ (0, 1] and v∈B λv = 1
Sums of this form are called convex combinations of the points in A.
We denote the space of linear functionals on Rn by (Rn )∨ . If ϕ is a
nonzero functional then any level set is a hyperplane and for c ∈ R the set
{x ∈ Rn |ϕ(x) ≤ c} is called a half space.
6
Definition 1.3. The intersection of a finite number of half spaces is called
a polyhedron.
The previous two definitions are related by the following theorem.
Theorem 1.4. A subset P ⊂ Rn is a polytope if and only if it is a bounded
polyhedron.
The proof can be found in [26] in chapter 1 and is harder than one would
expect. It is now immediately clear that the intersection of two polytopes is
again a polytope, which is not easy to prove directly from the definition.
If we would define a polytope as a bounded polyhedron then it would be
hard to prove that the Minkowski sum(defined in the following section) of
two polytopes would be a polytope, so theorem 1.4 is fundamental. 1
A subset F of a polyhedron P is called a face of P if F = ∅ or there
is a functional ϕ ∈ (Rn )∨ such that F = {x ∈ P |ϕ(x) = M }, where M
is the maximum of ϕ on P . The face F is called the supporting face of ϕ
and is denoted with P ϕ . Because M is the maximum of ϕ on P we have
F = P ∩ {x ∈ Rn |ϕ(x) ≥ M }, so F itself is a polyhedron. If P is a polytope
it follows from theorem 1.4 that a face is again a polytope.
A face of dimension 0 is called a vertex and a face of dimension dim(P )−1
is called a facet. A vertex contains only one point and we can identify the
vertex with the point it contains. The set of vertices is denoted with vert(P ).
The set of all faces of a polyhedron has the following structure.
Definition 1.5. A polyhedral complex is a collection C of polyhedra such
that
1. If P ∈ C then every face of P is an element of C.
2. If P, Q ∈ C then P ∩ Q is a face of P and of Q.
The support |C| of C is the union over all elements of C. We call C a
polyhedral subdivision of |C|. Notice that |C| is not necessarily a polyhedron.
Sometimes we refer to the elements of C as the cells of the complex. The
cells are partially ordered by inclusion. If all maximal cells have the same
dimension n we say that C is pure of dimension n and denote dim(C) = n.
The vertices of the cells of C are cells of C by property 1. and are called the
vertices of C. If C is pure of dimension n the cells of dimension n − 1 are
called the facets of C.
1
This version of theorem 1.4 gives no easy proof for the fact that the Minkowski sum
of two polyhedra is again polyhedron, which follows directly from a more precise version,
see page 30 of [26].
7
An isomorphism ϕ between complexes C and D is a bijection ϕ : |C| → |D|
which is affine on all elements of C and induces a bijection C → D.
Now we give an example of a polyhedral complex we encounter later.
Example 1.6. Let P ⊂ R × Rn be a polytope and let Pmax ⊂ P be the set
of all (λ, x) ∈ P such that if (t, x) ∈ P for some t ∈ R then t ≤ λ. Thus Pmax
consists of all points of P which are maximal on a line of the form R × {x}
with x ∈ Rn . The set of all faces of P contained in Pmax is called the upper
envelope of P . We can define the set Pmin as the set of points which are
minimal on lines of the form R × {x}, then the set of all faces contained in
Pmin is called the lower envelope of P .
Lemma 1.7. The upper envelope of a polytope P is a polyhedral subdivision
of Pmax and the lower envelope is a polyhedral subdivision of Pmin .
Proof. It follows directly from the definitions that the upper envelope is
a polyhedral complex. Now let p = (λ, x) ∈ Pmax an let H1 , ..., Hk be
the half spaces defining P . By definition of Pmax the must be an i such
that (λ + ε, x) ∈
/ Hi for every ε ∈ R>0 . Now p is contained in the face F
defined by Hi . If L is the boundary of Hi then F = L ∩ P . If w ∈ L then
w + (ε, 0, ..., 0) ∈
/ Hi for any ε ∈ R>0 , so F is an upper face of P .
The case of the lower envelope is identical.
1.2
The Newton polytope and its subdivisions
We will regularly see the following class of polytopes throughout this thesis.
The importance of the Newton polytope for tropical geometry is explained
by theorem 1.28 If K is a field the ring of Laurent polynomials is the ring
−1
−1
K[z
polynomial is a finite sum f =
P 1 , ..., znv, z1 , ..., zn ]. Thus a vLaurent
v1
a
z
,
with
a
∈
K
and
z
=
z
·
· · znvn . This gives the following
v
1
v∈Zk v
polytope.
P
Definition 1.8. If f = v∈Zn av zv is a Laurent polynomial then the Newton
polytope Newt(f ) is the convex hull of the set {v ∈ Zn |av 6= 0}.
We can immediately verify that Newt(f ) behaves nicely with respect to
multiplication. The Minkovski sum A+B of two subsets A, B ⊂ Rn is defined
by A + B = {a + b|a ∈ A, b ∈ B}.
Lemma 1.9. If f and g are Laurent polynomials over a field, then Newt(f g) =
Newt(f ) + Newt(g).
8
P
P
P
i
i
i
Proof.
If
f
=
a
z
and
g
=
b
z
,
then
f
g
=
i
i
i
i
i ci z , where ci =
P
n
j,k:j+k=i aj bk . If ci 6= 0 then there are j, k ∈ Z such that aj 6= 0 and
bk 6= 0, so i = j + k with j ∈ Newt(f ) and k ∈ Newt(g). Thus Newt(f g) ⊂
Newt(f ) + Newt(g).
To show the converse it is enough to show that all vertices of Newt(f ) +
Newt(g) are contained in Newt(f g). If k is a vertex of Newt(f ) + Newt(g),
then k is a sum of unique vertices i ∈ Newt(f ) and j ∈ Newt(g). Thus the
coefficient ck = ai bj 6= 0, so k ∈ Newt(f g).
The Newton polytope is an example of a lattice polytope, which is a
polytope such that all vertices are integral points. A polyhedral subdivision
of a lattice polytope is called a lattice subdivision if all of its elements are
lattice polytopes. It is clear that any lattice polytope can be realized as the
Newton polytope of a polynomial.
We now describe a way to make lattice subdivisions of a lattice polytope.
This method works similarly for arbitrary polytopes and can be found in
lecture 5 of [26].
Let Q ⊂ Rn be a lattice polytope and let A ⊂ Q be a set of integral
points containing all vertices of Q and let v : A → R be any function. In
the applications of this construction Q will be the Newton polytope of a
polynomial f and v will depend on the coefficients of f .
We define a polytope P ⊂ Rn+1 by P = conv{(v(i), i)|i ∈ A}. We have a
natural projection π : P → Q.
Lemma 1.10. The restriction π : Pmax → Q is a bijection.
P
Proof.
If
x
∈
P
then
we
can
write
x
=
i∈A λi (v(i), i) with λi ∈ [0, 1] and
P
P
λ
=
1,
so
π(x)
=
λ
i
∈
Q.
Thus
π(P ) ⊂ Q. It is clear that
i∈A i
i∈A i
π is injective. Now let x ∈ Q. We have Q =Pconv(A) because A contains
all vertices of Q. Thus P
we can write x =
i∈A λi i with λi ∈ [0, 1] and
P
i∈A λi (v(i), i) ∈ P . Thus Pmax must contain an
i∈A λi = 1. Now y =
element ỹ such that y1 = ỹ1 ,...,yn = ỹn . Clearly π(y) = π(ỹ) = x so π is
surjective.
If p is a vertex of a face contained in Pmax then p is also a vertex of
P so p is of the form (v(i), i1 , ..., in ) for some i ∈ A, so π(p) = (i1 , ..., in )
is an integral point. Thus the projection of the upper envelope defines a
lattice subdivision of Newt(f ). A subdivision of a polytope which can be
constructed in this way is called a regular subdivision.
We can immediately construct an interesting lattice subdivision.
9
Proposition 1.11. If Q ⊂ Rn is a lattice polytope there is a function v : Q ∩
Zn → Q such that the set of vertices of the regular subdivision induced by v
is Q ∩ Zn .
Proof. After a translation we can assume Q is contained in the box [0, N ]n
for some N ∈ N. Let f : [0, N ] → R be a concave function which is nowhere
linear, P
for example f (t) = N 2 − t2 . Now we define v : Q ∩ Zn → Q by
v(i) = nj=1 f (ij ) and note that the image indeed lies in Q.
We must show that (v(k), k) is an upper vertex of the polytope conv({(v(i), i)|i ∈
n
Q ∩ Zn }) in R × Rn for
P every k ∈ Q ∩ Z .
Suppose that k = i∈A λi i with A ⊂ Q ∩ Zn and λi ∈ (0, 1].
We get
v(k) =
n
X
j=1
n X
X
X
f(
λi ij ) ≥
λi f (ij )
j=1 i∈A
i∈A
=
X
i∈A
λi
n
X
j=1
f (ij ) =
X
λi v(i)
i∈A
by applying Jensen’s inequality to each term of the first sum. This shows
v(k) is maximal, so (v(k), k) lies in the upper envelope.
Because f is not linear we have equality if and only if ij = kj for all i ∈ A,
which can only happen if A = {k}. This shows (v(k), k) is not contained in
conv({(v(i), i)|i ∈ Q ∩ Zn − {k}}), so it is a vertex.
The function used in the proof of the previous lemma is generally not the
most convenient when constructing a subdivision. In the following example
we use a simpler one.
Example 1.12. Consider the ring C(t)[u, w] of polynomials in two variables
over the field of rational functions over C and let
f = u3 + w3 + 1 + u2 w3 + u3 w2 + tu + tu2 + tw + tw2 + tu3 w
+tuw3 + t2 uw + t2 u2 w + t2 uw2 + t2 u2 w2 ,
then Newt(f ) = conv({(0, 0), (0, 3), (3, 0), (3, 2), (2, 3)}). We define v : Newt(f )∩
Z2 → R by v(i) = valt (ai ) where ai is the coefficient of f at ui1 wi2 . It is easy
to check that figure 1.1 is the associated subdivision of Newt(f ).
10
Figure 1.1: The subdivision of Newt(f ) of example 1.12.
Not every lattice subdivision of a lattice polytope is regular, see [26] page
132 for a simple counterexample. Many results about regular subdivisions
can be found in [8]. More counterexamples can be found in [12].
1.3
Tropical algebra
Tropical varieties are also derived from algebra, so we give the necessary definitions of tropical algebra in this section. Similarly to algebraic geometry
there are two ways to define tropical polynomials, as abstract sums of monomials or as polynomial functions. As in the case of finite fields a distinct
formal sums can define the same functions.
In this thesis we only consider tropical hypersurfaces which are easy to
define directly using a function on Rn and in this chapter this concrete definition would be sufficient. In chapter 3.2 briefly consider the algebra of tropical
polynomials.
Definition 1.13. A semiring is a set R with addition ⊕ and multiplication
such that
• (R, ⊕) is a commutative monoid with identity element e⊕
• (R, ) is a monoid with identity element e
and for all a, b, c ∈ R the operations must satisfy
• a (b ⊕ c) = a b ⊕ a c
• (b ⊕ c) a = b a ⊕ c a
11
• e⊕ a = e⊕ .
The axioms are the same as those of a ring except for the lack of an
additive inverse and the fact that e⊕ a = e⊕ does not follow from the other
axioms.
A standard example is R≥0 with addition and multiplication. The following example is less intuitive.
Example 1.14. For t ∈ R>1 let Rt = R ∪ {−∞}, let x ⊕t y = logt (tx + ty )
and x t y = x + y, where we conveniently put t−∞ = 0. We have logt (x) ⊕t
logt (y) = logt (x + y) and logt (x) t logt (y) = logt (xy) for every x, y ∈ R≥0 ,
so the operation on Rt is just the operation of R≥0 transfered by the map
logt . Thus Rt is a semiring and Rt ∼
= R≥0 .
Definition 1.15. The tropical semiring is the set T = R ∪ {−∞} with
operations x ⊕ y = max{x, y} and x y = x + y.
It is not hard to check that T satisfies all the axioms of a semiring.
We can also get T as a limit of the semirings Rt given in the previous
example. We have max{x, y} = limt→∞ logt (tx + ty ). To see this let x =
max{x, y}, so log(1 + ty−x ) ≤ log(2) is bounded. Now limt→∞ (tx + ty ) =
y−x )
= x. This gives an immediate proof that T is a semiring.
x+limt→∞ log(1+t
log(t)
The process of taking the limit of the semirings Rt is called Maslov dequantization. It is explained briefly on page 28 of [14].
L
Definition 1.16. A tropical polynomial is a formal sum i∈I ai xi11 ...xinn
where I ⊂ Zn is a finite index set, ai ∈ T and x1 , ..., xn are real variables.
If x = (x1 , ..., xn ) and i = (i1 , ..., in ) ∈ Zn we use xi instead of xi11 ...xinn
and i · x = i1 x1 + ... + in xn , where · is the standard inner product.
L
Using the definition of ⊕ and a tropical polynomial f = i∈I ai xi
defines a function on Rn given by f (x1 , ..., xn ) = maxi∈I {ai + i · x}. We say
two tropical polynomials are equivalent if they define the same function.
Example 1.17. Two tropical polynomials with different coefficients can be
equal. For example 0 ⊕ x ⊕ x2 and 0 ⊕ x2 both define the function f (x) =
max{0, 2x} on R.
For every equivalence class of tropical polynomials we can define a pick
canonical polynomial with the least possible number of terms as follows. If f
is given by maxi∈I {ai +i·x} and k ∈ I define Ck = {x ∈ Rn |ak +k·x = f (x)}.
On Ck the function f is given by ak + k · x.
Put J = {i ∈ I|Ci◦ 6= ∅} and fred = maxi∈J {ai +i·x} then f and fred define
the same function on Rn . It is clear that all tropical polynomials equivalent
to f have the same reduced polynomial.
12
1.4
Tropical hypersurfaces
Analogous to algebraic geometry we want to assign a hypersurface to a tropical polynomial. The following definition is natural.
Definition 1.18. If f is a tropical polynomial in n variables with at least 2
terms then the hypersurface V (f ) of f is the set of all points of Rn where f
is not locally linear.
It is clear that equivalent tropical polynomials define the same tropical
hypersurface. A tropical hypersurface in R2 is called a tropical curve.
Lemma 1.19. For a tropical polynomial f = maxi∈I {ai + i · x} the following
sets are equal
1. The hypersurface V (f ).
2. The set of all p ∈ Rn where f achieves its maximum at least twice.
T
3. The intersection k∈I (Ck◦ )c .
Proof. Suppose f is not locally linear in p ∈ Rn , then there are distinct
k1 , k2 ∈ I such that f is equal to ai + ki · x on points arbitrarily close to p.
Because f is continuous it follows that f (p) = a1 + k1 · p = a2 + k2 · p, so
the first set is contained in the second.
If f achieves its maximum for both k1 , k2 ∈ I it is clear that p cannot lie
in Ck◦ for any k ∈TI, so the second set is contained in the third.
Suppose p ∈ k∈I (Ck◦ )c , then there must be distinct k1 , k2 ∈ I such that
a1 + k1 · p = a2 + k2 · p = f (p). If p ∈
/ V (f ) then a1 + k1 · x = a2 + k2 · x on
an open neighborhood of p, so k1 = k2 . This shows the third set is contained
in the first.
Example 1.20. Figure 1.20 is the tropical curve defined by max{0, 3y, 3x, 3x+
2y, 2x + 3y, x + 1, 2x + 1, y + 1, 2y + 1, 3x + y + 1, x + 3y + 1, x + y + 2, x +
2y + 2, 2x + y + 2, 2x + 2y + 2}. The edges which cross the axes extend to
infinity. If we consider figure 1.20 and 1.1 as graphs they are dual. We will
see that when we put the structure of a polyhedral complex on V (f ) they
can be seen as dual polyhedral complexes.
We now construct a polyhedral subdivision of Rn such that the union over
all facets is V (f ). In the case of figure 1.2 it is easy to see that this is possible.
This way we also get a polyhedral subdivision of V (f ).
For A ⊆ I let CA = {x ∈ Rn |ai + i · x = f (x) for all i ∈ A}. Notice that
C{k} = Ck and CA = ∩k∈A Ck . Let V(f ) = {CA |A ⊂ I and #A ≥ 1} ∪ {∅}.
13
Figure 1.2: The tropical curve of example 1.20.
Proposition 1.21. For a tropical polynomial f = maxi∈I {ai + i · x} the
collection V(f ) is a polyhedral complex and V(f ) = V(fred ).
Proof. If fred = maxi∈J {ai + i · x}, then for any A ⊆ J the set CA is the same
in V(f ) as in V(fred ) as f and fred define the same function on Rn . Thus
V(fred ) ⊆ V(f ).
For all A ⊆ I have CA = ∩k∈A Ck and Ck = {x ∈ Rn |ak + k · x = f (x)} is
a polyhedron, so all sets in V(f ) are polyhedra.
We now want to show that CA ∩ CB = CA∪B is a face of CA and CB
for any nonempty A, B ⊂ I. Let k, j be distinct elements of I then the
equation ak + k · x = aj + j · x defines a hypersurface H such that Ck and
Cj lie in different half spaces defined by H. By definition of H we have
H ∩ Ck = H ∩ Cj and Cj ∩ Ck ⊂ H, so Cj ∩ Ck = H ∩ Ck = H ∩ Cj is
a face of both Cj and Ck . For any nonempty A ⊂ I and k ∈ A we have
CA ∩ Ck = ∩i∈A Ci ∩ Ck , so CA is an intersection of faces of Ck , so CA is also a
face of Ck . By the same argument CA ∩ CB = CA∪B is a face of Ck for some
k ∈ A, so CA ∩ CB is also a face of CA .
Now let j ∈ J, so Ck◦ 6= ∅ and dim(Cj ) = n. The boundary of Cj is
contained in the union of the hypersurfaces Hi defined by aj + j · x = ai + i · x
for j 6= k. Each Hj defines a face Hi ∩ Cj = C{i,j} of Cj of dimension at most
n = 1 so these faces are proper.
If F is a facet of Cj then dim(F ) = n − 1 and F = ∪i6=j Hi ∩ F . We
have dim(Hi ∩ F ) ≤ dim(F ) and the union is finite so there is a k 6= j
with dim(Hk ∩ F ) = dim(F ). As Hk ∩ F = (Hk ∩ Cj ) ∩ F it follows that
14
Hj ∩ F is a face of F so F = F ∩ Hj . Therefore F ⊆ Hk ∩ Cj , so F is
also a face of Hk ∩ Cj = C{k,j} . As dim(F ) = dim(Hk ∩ F ) this implies
F = C{k,j} ∈ V(fred ).
It now follows directly that any face of Cj is also in V(fred ) because it is
the intersection of all facets containing it. The same statement also follows
for faces of general CA ∈ V(fred ), because a face of CA is also a face of Ck for
any k ∈ A. This shows V(fred ) is a polyhedral complex.
Now pick a general CA ∈ V(f ) then CA = ∪j∈J CA ∩ Cj . As CA ∩ Cj is a
face of CA it follows by comparing the dimensions that the is a j ∈ J such
that CA = CA ∩ Cj . As CA ∩ Cj is a face of Cj and V(fred ) is a polyhedral
complex it follows that CA ∈ V(fred ), so it follows that V(f ) = V(fred ).
Corollary. The complex V(f ) is pure of dimension n and any nonempty
element is the intersection of maximal elements. The tropical curve V (f ) is
the union over all facets of V(f ).
Proof. The first statement is clear for V(fred ), so it also holds for V(f ). The
second part follows directly from lemma 1.19.
1.5
The Newton polytope of a tropical curve
Now we give a different construction of V(f ). This construction allows us to
make the connection with subdivisions mentioned in example 1.20. To make
this subdivision we need to define what the Newton polytope of a tropical
polynomial is and we need to introduce normal cones.
Definition 1.22. For a polytope P with face F the normal cone NF (P ) of
P at F is the subset of (Rn )∨ defined by
NF (P ) = {ϕ ∈ (Rn )∨ |ϕ(z) ≤ ϕ(y) for all z ∈ P and y ∈ F }.
This means that NF (P ) consists of all functions ϕ such that the maximum
of ϕ on P is attained on all of F . If ϕ is a functional such that P ϕ = F , then
ϕ ∈ NG (P ) for any face G with G ⊆ F and F is the maximal face for which
ϕ is contained in the normal cone.
It is usual to identify Rn with its dual (Rn )∨ via the map given by x 7→
(y 7→ x · y), where · is the usual inner product. With this identification the
normal cone is given by
NF (P ) = {x ∈ Rn |x · z ≤ x · y for all z ∈ P and y ∈ F }.
15
We only need to check maximality of ϕ on vertices so we have
NF (P ) = {x ∈ Rn |x · z ≤ x · y for all z ∈ A and y ∈ vert(F )},
(1.5.1)
where A is any subset of P which contains vert(P ).
The following properties of normal cones can be found in [26]. A polyghedron C is a cone if λx ∈ C for all x ∈ C and λ ∈ R≥0 .
Proposition 1.23. Let P ⊂ Rn be a polytope and let F and G be nonempty
faces of P .
1. The normal cone NF (P ) is a polyhedron and a cone.
2. We have NF (P ) ⊂ NG (P ) if and only if G ⊂ F and equality if and
only if F = G.
3. The normal cone satisfies dim(F ) + dim(NP (F )) = n.
4. If F1 , ..., Fn are faces of P then there is a face K such that ∩ni=1 NFi (P ) =
NK (P ).
We can use normal cones to find the upper faces of a polytope. Let
P ⊂ R × Rn be a polytope and let H1 = {1} × Rn .
Proposition 1.24. A face F of P is an upper face if and only if NF (P ) ∩
H1 6= ∅.
Proof. If we consider (t, x) ∈ R × Rn as a functional it defines a face P by
F = {(s, y) ∈ R × Rn |ts + x · y = M }, where M is the maximum of (t, x) on
P . If t > 0 and (s, y) ∈ P (t,x) then (s, y) is a maximal point of P on the line
R × y as M is the maximum of (t, x) on P . Thus P (t,x) is an upper face.
If NF (P ) ∩ H1 6= ∅ pick (1, x) ∈ NF (P ) ∩ H1 , then F ⊆ P (1,x) which is
a upper face, so F is an upper face as well. This shows one direction of the
equivalence.
Let H1 , ..., Hn be the half spaces defining P and let L1 , ...Ln be the corresponding hyperspaces, then P ∩ Li is a face of P for i = 1, ..., n and the
boundary of P is contained in ∪ni=1 Li .
Let F be a nonempty upper face of P then F = ∪ni=1 F ∩ Li and F ∩ Li
is a face of F . We reorder the half spaces such that L1 ∩ F, ..., Lk ∩ F are
proper faces of F and Lk ∩ F, ..., Ln ∩ F are equal to F . As the dimension of
a proper face of F is smaller that the dimension of F it follows that k < n.
Let z be a point of F which is not contained in any proper face of F .
If Li ∩ F is a proper face of F then z ∈ Hi◦ . Thus z is contained in the
16
interior of ∩ki=1 Li , so there is an ε > 0 such that z + (ε, 0, ..., 0) ∈ ∩ki=1 Li . As
z + (ε, 0, ..., 0) ∈
/ P there must be a j > k such that z + (ε, 0, ..., 0) ∈
/ Hj .
The half space Hj is defined by a functional (t, x) and a constant M ∈ R
via Hj = {(s, y) ∈ R × Rn |ts + x · y ≤ M }. As w · (t, x) = M and
(z + (ε, 0, ..., 0)) · (t, x) > M it follows that t > 0. Now (1, 1t x) also defines
Hj , so (1, 1t x) is maximal on F . Thus (1, 1t x) is the required element of
NF (P ) ∩ H1 .
We will later combine the previous proposition with the following general
fact about cones.
Lemma 1.25. Let C ⊂ Rn be a polyhedron which is a cone of dimension
d and let H be an affine hyperspace which does not contain the origin. If
C ∩ H 6= ∅ then dim(C ∩ H) = d − 1.
Proof. The smallest linear subspace L of Rn which contains C has dimension
d and H ∩ L is a hyperspace in L, so without loss of generality we can assume
dim(C) = n.
After a linear transformation we can assume H is defined by x1 = 1. If all
interior points of C satisfy x1 ≤ 0 then all points of C satisfy x1 ≤ 0, which
is a contradiction because C ∩ H = ∅ . Thus we can pick a a point x ∈ C ◦ .
After multiplication with a positive constant it follows that H must contain
an interior point of C, so C ∩ H contains a disc of dimension d − 1, which
proves the lemma.
Now we can define the Newton polytope of a tropical polynomial and
check that this definition makes sense.
Definition 1.26. If f = maxi∈I {ai +i·x} is a tropical polynomial Newt(f ) =
conv(I).
Lemma 1.27. The Newton polytope of f only depends on the function defined
by f on Rn
Proof. It is enough to show Newt(f ) = Newt(fred ) and it is clear that
Newt(fred ) ⊆ Newt(f ).
Let k be a vertex of Newt(f ), then Nk (Newt(f )) has an interior point x.
This point satisfies i · x < k · x for every i ∈ I − {k}, so for a sufficiently
large λ ∈ R we have i · λx + ai < k · λx + ak for every i ∈ I − {k}. Thus
f is equal to k · x + ak on λx + Nk (Newt(f )). As λx + Nk (Newt(f )) is a
translated cone of dimension n it follows that Ck◦ 6= ∅, so k · x + ak is a term
of fred and k ∈ Newt(fred ). This shows Newt(f ) = Newt(fred ).
17
We can now use the construction from section 2 to define a lattice subdivision of Newt(f ). We have seen that the function v : I → R given by v(i) = ai
defines a polytope P = conv{(ai , i)|i ∈ I} ⊂ R×Rn and the projection of the
upper hull of P is lattice subdivision of Newt(f ). We denote this subdivision
with S(f ).
The following elementary proof is based on lecture notes by Bernd Sturmfels which can be found at [24] A proof using the Legendre transform can be
found in [14].
Theorem 1.28. If f is a tropical polynomial there is an inclusion reversing
bijection between the nonempty cells of V(f ) and the nonempty cells of S(f ).
Proof. By definition we have an inclusion preserving bijection between S(f )
and the upper hull of P . We can include V(f ) in R × Rn as {1} × V(f ), so
it suffices to give an inclusion reversing bijection Φ from the upper hull of P
to {1} × V(f ).
For an upper face F of P define Φ(F ) = NF (P ) ∩ H1 , where H1 = |{1} ×
V(f )| = {1} × Rn . Because any vertex of P is of the form (ai , i) with i ∈ I
formula 1.5.1 shows the normal cone of a face F is given by
NF (P ) = {(t, x) ∈ R × Rn |tai + x · i ≤ taj + x · j for all i ∈ I, j ∈ A},
where A is the set of all j ∈ I such that (aj , j) is a vertex of F . Thus
NF (P ) ∩ H1 = {1} × CA which is a cell of {1} × V(f ). In particular N(ai ,i) =
{1} × Ci for all i ∈ I such that (ai , i) is a vertex.
The function Φ is inclusion reversing by lemma 1.23.
Let k ∈ I such that Ck◦ 6= ∅, pick x ∈ Ck◦ and let F = P (1,x) then (1, x) ∈
NF (P ) so F is a upper face by lemma 1.24. Because NF (P ) ∩ H1 is a cell of
{1} × V(f ) and {1} × Ck is maximal it follows that NF (P ) ∩ H1 = {1} × Ck .
Now lemma 1.25 shows that F is a vertex of P and it is clear that F can
only be (ak , k).
Let {1} × CA be any nonempty cell of {1} × V(f ) then we can choose
A such that Ck◦ 6= ∅ for all k ∈ A by corollary 1.4. Now {1} × CA =
∩k∈A N(ak ,k) (P ) ∩ H1 = (∩k∈A N(ak ,k) (P )) ∩ H1 . By lemma 1.23 there is a face
F of P such that NF (P ) = ∩k∈A N(ak ,k) (P ) and F is an upper face by lemma
1.24. Thus Φ is surjective.
If F and G are upper faces of P and NF (P ) ∩ H1 = NG (P ) ∩ H1 then
(NF (P )∩NG (P ))∩H1 = NF (P )∩H1 By proposition 1.23 there is a face K of
P with NF (P ) ∩ NG (P ) = NK (P ) which contains NF (P ) and NG (P ) and by
lemma 1.24 this is an upper face. By lemma 1.25 the dimensions of NF (P ),
NG (P ) and NK (P ) are equal, so by proposition 1.23 the dimensions of F , G
and K are equal. As F and G are faces of K it follows that F = G = K.
This shows the assignment Φ is an inclusion reversing bijection.
18
Chapter 2
The amoeba of a planar curve
In this chapter we give the definition of an amoeba which was first introduced
in [8]. The idea of applying a logarithmic function to an algebraic variety is
not a completely unexpected one. Applying the logarithm to a real algebraic
curve was explored by Oleg Viro, see for example [19]
Also a basic result of multivariable complex analysis is that there is a
convergent Laurent series on a set of complex numbers if and only if the
logarithm of the absolute values of this set is a convex set. This result is
used to derive basic results about amoebas and we cite it as proposition 2.5.
We will give some elementary properties and examples of amoebas. The
picture are drawn with the algorithm in appendix 6.
2.1
Definitions and examples
In this chapter we work with Laurent polynomials, which are elements of
K[z1 , ..., zn , z1−1 , ..., zn−1 ], where K is a field. In most of the section C is the
base field, n = 2 and we use u and w instead of z1 and z2 . Throughout this
chapter f (z) = f (z1 , ..., zn ) denotes a Laurent polynomial, and Cf denotes
the hypersurface defined by f in (C∗ )n .
If K is a field then an absolute value on K is a function | · |K : K → R≥0
such that for all a, b ∈ K we have
1. |a|K = 0 if and only if a = 0
2. |ab|K = |a|K |b|K
3. |a + b|K ≤ |a|K + |b|K
If we have the stronger condition
19
3’ |a + b|K ≤ max{|a|K , |b|K }
then | · |K is called non-archimedean.
If K has an absolute value then we have a map LogK : (K ∗ )n → Rn given
by LogK (x1 , ..., xn ) = (log(|x1 |K ), ..., log(|xn |K )). If K = C we use | · | instead
of | · |C and Log instead of LogC . In this chapter we only consider amoebas
over the complex numbers and in the next chapter we look at amoebas over
a field with a non-archimedean absolute value.
Definition 2.1. If X ⊂ (K ∗ )n is an algebraic variety then the amoeba of X
is the set LogK (X).
We can immediately note the following basic properties.
Proposition 2.2. If X is an irreducible complex variety then the amoeba of
X is closed and connected.
Proof. The absolute value map is a proper map by the Heine-Borel theorem
and (C∗ )n is locally compact so the absolute value map is a closed map. Thus
the logC map is also closed, so the amoeba of X is closed.
An irreducible complex variety is always connected(see for example section
7.2 of [22]), so the amoeba is connected as well.
It is not immediately clear that this definition gives an interesting set.
However the following picture, which shows the amoeba of the curve given
by f (u, w) = u3 + w3 + 2uw + 1 clearly shows that the amoeba is an object
with non-trivial geometry.
Figure 2.1: The amoeba of the curve defined by f (u, w) = u3 + w3 + 2uw + 1.
20
Example 2.3. The amoeba of an affine hyperplane. In (C∗ )2 such a hyperplane is given by an equation of the form f (u, w) = au + bw + c with
a, b, c ∈ C where at least two of a, b, c are not zero.
so the amoeba is the set {(x, log(|c|) − log(|b|)) : x ∈
If a = 0 then w = −c
b
R}, which is the line defined by y = log(|c|) − log(|b|) in R2 . The case b = 0
is similar.
If c = 0 then the amoeba of f is the line given by y = x+log(|a|)−log(|b|).
Figure 2.2: The boundary of the amoeba given by f (u, w) = u + w + 1.
In the case a, b, c are not zero, then we can consider the case a = b = c = 1,
as other values give a translation of this amoeba by (log(c) − log(a), log(c) −
log(b)). In this case Cf = {(u, −u − 1) : u ∈ C∗ }. If |u| > 1 then | − u − 1| =
|u + 1| can assume any value in [|u| − 1, |u| + 1], if |u| = 1, then |u + 1| can
assume any value in (0, 2] and if 0 < |u| < 1 then |u + 1| can assume any
value in [1 − |u|, |u| + 1]. Thus the absolute values of the points in Cf is the
part of R2>0 bounded by the lines {(t, t + 1)|t ∈ (0, ∞)}, {(t, 1 − t)|t ∈ (0, 1)}
and {(t, t − 1)|t ∈ (1, ∞)}. We get the amoeba by applying the logarithm.
Notice that in the cases where the area of the amoeba of a line is 0 the
Newton polytope of f is not of full dimension.
2.2
Laurent series
From now on we only consider the case of the amoeba of a hypersurface defined over the complex numbers by a Laurent polynomial f . To understand
21
the complement of the amoeba we need to look at the Laurent
P series expansion of f1 . A Laurent series S centered at 0 is a formal sum v∈Zn av zv , where
av ∈ C and zv = z1v1 · · · znvn . Because Zn has no natural order a Laurent series is said to converge in a point if it converges absolutely. The domain of
convergence is the interior of the set of all point where S converges.
Example 2.4. Let f (u, w) = u + w + 1Pbe the polynomial
example
iform
j
2.3 and consider the Laurent series S = i,j∈Z2 (−1)i+j i+j
u
w
,
where
we
j
i+j
note that j = 0 if i < 0 or j < 0. If S converges in (u0 , w0 ) then we
P
k
k
have S(u0 , w0 ) = ∞
k=0 (−1) (u0 + w0 ) by the binomial theorem. This is a
geometric series so it follows that S is a Laurent expansion of f1 . .
Applying the binomial theorem to the absolute value gives
∞
X i + j X
X
i+j i + j
i j
i
j
(|u| + |w|)n ,
uw |=
|u| |w| =
|(−1)
j
j
2
2
n=0
i,j∈Z
i,j∈Z
so this series converges absolutely if and only if |u|+|w| < 1. Thus the domain
of convergence of S is the set {(u, w) ∈ (C∗ )2 ||u| + |w| < 1} and applying the
Log function to this domain gives one component of the complement of the
amoeba of f .
Using Pascals identity for binomial coefficients and similar arguments to
the previous example one can show that
X
X
−i+1 −i + 1
i j
−j+1 −j + 1
(−1)
u w and
(−1)
ui wj
j
i
2
2
i,j∈Z
i,j∈Z
are Laurent expansions of f1 and that the domains of convergence are {(u, w) ∈
(C∗ )2 ||w| < |u| − 1} and {(u, w) ∈ (C∗ )2 ||u| < |w| − 1}. Thus the domains of
convergence give the other components of the complement of the amoeba. A
consistent way to find certain Laurent expansions of f1 is given in the remark
at the end of this section.
The correspondence between components of amoebas and Laurent series
follows from a general result in multivariable complex analysis. A slightly
different version of this proposition is cited in [8]. See [17], theorem 2, for a
proof of part (b) and [11] theorem 2.3.2 for part (a).
P
Proposition 2.5. (a) If S(z) = v∈Zn av zv is a Laurent series centered at
0 the domain of convergence is of the form Log−1 (B) where B ⊂ Rn is a
convex open subset. 1
1
In multivaliable complex analysis such a domain is referred to as a logarithmically
convex Reinhardt domain.
22
(b) If ϕ(z) is a holomorphic function on a domain D of the form Log−1 (B)
with B ⊂ Rn open and connected then there is a unique Laurent series centered at 0 which converges to ϕ(z) on D.
We can directly apply this to amoebas.
Corollary. If f is a Laurent polynomial then the components of Rn −Log(Cf )
are convex and the components correspond bijectively to the Laurent series
expansions of f1 centered at 0.
Proof. Let A be a component of Rn − Log(Cf ) then f1 is a holomorphic
function defined on Log−1 (A). By part (b) of proposition 2.5 there is a
unique Laurent series S which converges to f1 on Log−1 (A). By part (a) of
the same proposition the domain of convergence of S is of the form Log−1 (B)
with B ⊂ Rn open and convex. We clearly have A ⊂ B and as B is convex
it is also connected, so A = B.
Thus the components of the complement of the amoeba are convex and
we can associate a unique Laurent series to each.
If S 0 is another Laurent series of f1 centered at 0, then it has a domain of
convergence of the form Log−1 (B) where B ⊂ Rn is a convex open subset.
As B cannot intersect the amoeba there is a component A of the complement
such that A ∩ B 6= ∅. If S is the power series expansion corresponding to A
by the first part of the proof then S 0 = S by part (b) of proposition 2.5.
Remark. It is possible compute some of the Laurent series of f1 directly. If
γ is a vertex
of Newt(f ) then we can write f (z) = aγ zγ (1 + g(z)), where
P
g(z) = v∈Zk ,v6=γ aaγv zv−γ . Using the geometric series we get a formal identity
∞
X
1
−γ
z
(−1)i g(z)i .
= a−1
γ
f (z)
i=0
(2.2.1)
Expanding the terms gives a Laurent series which converges on a component
of the complement of the amoeba. The details can be found in [8] or [7].
2.3
The amoeba and the Newton polytope
In this section we show some elementary properties of amoebas. In particular
all amoebas will look somewhat similar to figure 2.1, with a finite number of
tentacles. Furthermore the directions of the tentacles are prescribed by the
Newton polytope.
23
For proposition 2.7 we only consider curves in (C∗ )2 . This eliminates the
need to use Laurent series. The proofs in [8] use the Laurent series of f1 and
work in all dimensions.
For a vertex γ of Q = Newt(f ) the normal cone Nγ (Q) can be identified
with a subset of Rn by formula 1.5.1, so Nγ (Q) = {a ∈ Rn |a · (v − γ) ≤
0 for every v ∈ Q}.
Proposition 2.6. There is a vector b ∈ Nγ (Q) such that b + Nγ (Q) does not
intersect the amoeba of f .
Proof. For any M ∈ R we can find a vector b ∈ Nγ (Q) such that b·(v−γ) < M
for every v ∈ Q ∩ Zn different from γ. Now this condition also holds for every
element of b + Nγ (Q).
For any v ∈ (Q − {γ}) ∩ Zn and z ∈ (C∗ )n such that Log(z) ∈ b + Nγ (Q),
we have Log(|zv−γ |) = Log(z) · (v − γ) < M . This gives |zv | < |zγ |eM . Let
k + 1 be the number of nonzero coefficients of f , then
X
X
|av ||zγ |eM ≤ k max{|av | : v 6= 0}eM |zγ |.
av z v | <
|
v6=γ
v6=γ
M
This
P showsv that if weγ pick M such that k max{av : v 6= γ}e ≤ |aγ | we get
| v6=γ av z | < |aγ ||z |. Thus f cannot have a zero in z, so a does not lie in
the amoeba.
Remark. As stated in the proof it is enough to find b such that (b, ω − γ) <
M for every integral ω ∈ Q different from γ, where M ∈ R is such that
k max{|av | : v 6= 0}eM ≤ |aγ | where k+1 is the number of nonzero coefficients
of f .
Remark. If we compute the cones in the example 2.3 we can use M = log( 12 )
so we can pick b(0,0) = (log( 21 ), log( 12 )), b(1,0) = and b(0,1) = (0, log( 12 )).
Proposition 2.7. The translated cones b+Nγ (Q) in the previous proposition
lie in distinct components of the complement of the amoeba.
Proof. Let γ and δ be vertices of Newt(f ) and let bγ + Nγ (Q) and bδ + Nδ (Q)
be the cones which do not intersect the amoeba of f . Also let Aγ and Aδ be
the components containing bγ + Nγ (Q) and bδ + Nδ (Q).
First we assume γ and δ are adjacent vertices. We can assume that γ and
δ lie on the y-axis, Q lies in R2≥0 and Q meets the x-axis. By proposition
5.6 this can be achieved by an automorphism of C[u, w, u−1 , w−1 ] defined in
equation 5.1.1 and by noting that Q can be moved by integer vectors without
changing the curve in (C∗ )2 . Now f does not contain negative exponents and
is not divisible by u and w in C[u, w]. In this case f also defines a curve in
24
Figure 2.3: The boundary of the amoeba given by f (u, w) = u + w + 1 with
the translated normal cones.
C2 which restricts to Cf in (C∗ )2 . On the w-axis of C2 the polynomial f is
given by f (0, w) which has more than one term, so f has a zero a = (0, a2 )
with a2 6= 0.
As curves have no isolated points there are points {(aj1 , aj2 )}j∈N ∈ Cf such
that limj→∞ (aj1 , aj2 ) = a. Now limj→∞ log(|(aj1 )|) = −∞ and limj→∞ log(|aj2 |) =
log(|(a2 )|). Thus for sufficiently large j the point (aj1 , aj2 ) lies between
bγ + Nγ (Q) and bδ + Nδ (Q). As components are convex this shows bγ + Nγ (Q)
and bδ + Nδ (Q) cannot lie in the same component so Aγ 6= Aδ .
Now we assume γ and δ are arbitrary vertices with Aγ = Aδ and we show
that we can find an adjacent vertex η of γ with Aγ = Aη . We note that by
convexity Aγ contains bγ + bδ + Nγ (Q) + Nδ (Q).
We can write Nγ (Q) = H1 ∩H2 for open half planes H1 and H2 . If Nδ (Q)∩
H1 = ∅ = Nδ (Q) ∩ H2 then Nγ (Q) + Nδ (Q) = R2 , which is impossible, so we
can assume Nδ (Q) ∩ H1 6= ∅. If η is the adjacent vertex of γ for which Nη (Q)
lies in H1 then any translation of Nη (Q) intersects bγ + bδ + Nγ (Q) + Nδ (Q).
Thus Aγ ∩ Aη 6= ∅ and therefore Aγ = Aη , which completes the proof.
25
2.4
The components of the complement of
the amoeba
Using the integral formula 2 for the coefficients of the Laurent expansion
of f1 it is possible to derive more information about the complement of the
amoeba. In [7] the authors construct a distinct point in Newt(f ) ∩ Zn for
every component of Acf . The same function was constructed by Mikhalkin
in [15] using the topological linking number.
The results can be stated as follows.
Theorem 2.8 (Fosberg, Passare, Tsikh). The order of a component of Acf
is an element of Newt(f ) with integer coefficients and the orders of distinct
components are distinct.
Corollary. For a Laurent polynomial f the number of distinct Laurent expansions of f1 is at most the number points in Newt(f ) ∩ Zn and at least the
number of vertices.
The following property of the order will be useful later on.
Proposition 2.9. P
Let k ∈ Zn ∩Newt(f ) and let z ∈ (C∗ )n such that Log(z) ∈
/
j
c
k
Af and |ak z | > | j6=k aj z |, then the order of the complement of Af that
contains Log(z) is k.
2
See for example [11] theorem 2.7.1
26
Chapter 3
Puiseux series and Kapranov’s
theorem
The aim of this chapter is to give a proof of Kapranov’s theorem. This theorem describes the amoeba of a hypersurface over an algebraically closed field
with a non archimedean absolute value in terms of tropical geometry. It was
proved by Mikhail Kapranov in an unpublished manuscript [10]. Kapranov,
Einslieder and Lind later published a significantly more abstract version of
the theorem in [4].
The original result is quoted in [14] and a more general result is proved
in the forthcoming book [13] by Maclagan and Sturmfels. The proof in this
chapter is based on course notes [24] by Bernd Stumfels.
3.1
Valuations and Puiseux series
Let K be a field with absolute value | · |K . The absolute value | · |K is called
non-Archimedean if we have |a + b|K ≤ max{|a|K , |b|K } for every a, b ∈ K.
This is a stronger version of the normal triangle inequality. It is often intuitive
to define non-Archimedean absolute values in terms of valuations.
Definition 3.1. A valuation on a field K is a function val : K → R ∪ {∞}
such that for all a, b ∈ K we have
1. val(a) = ∞ if and only if a = 0
2. val(ab) = val(a) + val(b)
3. val(a + b) ≥ min{val(a), val(b)}
It follows directly from the seconds axiom that val(1) = 0 and val(a−1 ) =
− val(a). We also note the following.
27
Lemma 3.2. If val(a) 6= val(b) then val(a + b) = min{val(a), val(b)}.
Proof. We can assume val(a) < val(b), then val(a+b) ≥ val(a). Suppose that
val(a + b) > val(a), then val(a) = val((a + b) − b) ≥ min{val(a + b), val(b)} >
val(a) which is impossible, so val(a + b) ≤ val(a).
We can switch between non-Archimedean absolute values and valuations
by the following proposition.
Proposition 3.3. If | · |K is a non-Archimedean absolute value on a field
K then val(a) = − log |a|K defines a valuation on K. Conversely if val
is a valuation on K the function defined by |a|K = e− val(a) defines a nonArchimedean absolute value on K.
Now we construct the field of Puiseux series, which is an extension of the
field of Laurent series in which more exponents are allowed.
The valuation
P on ithe field C((t)) of formal Laurent series over C defined
by for p =
i∈I ai t by val(p) = min{i ∈ I|ai 6= 0} is an example of a
non-Archimedean valuation. For k > 0 the field C((t1/k )) is an extension of
C((t)) of degree k and the valuation can be extendedPby the same definition.
A Puiseux series over C is a series of the form i∈I ai ti , where ai ∈ C
and I ⊂ Q is a set of fractions with a common denominator and a minimal
element. We denote the set of Puiseux
series over C with C{{t}}. If k is a
P
common denominator for I then i∈I ai ti ∈ C((t1/k )). This shows we have
S
1
1
k
C((t k )) ⊂ C{{t}} for all k ∈ Z>0 and C{{t}} = ∞
k=1 C((t )). Thus C{{t}}
is a field extending
C((t)).
P
i
If p = i∈I ai t is a Puiseux series we put val(p) = min{J}. It is easy to
verify that val is a non-Archimedean valuation on C{{t}}.
The following theorem is due to Puiseux but was also known to Newton.
In the language of modern algebra this theorem can be proved easily by
applying Hensel’s lemma to the ring C[[t]] as is done in [6].
Theorem 3.4 (Puiseux). The field C{{t}} is the algebraic closure of C((t)).
We can now take a look at amoebas defined over K = C{{t}}. The
absolute value on K is given by |p|K = e− val(p) , so the logK function on
(K ∗ )n is given by logK (p1 , ..., pn ) = (− val(p1 ), ..., − val(pn )).
We now compute a very simple example of an amoeba which is similar to
the general case.
Example 3.5. Let Cf be the curve in (K ∗ )2 defined by f = u + w + t and
let T be the tropical curve V (x ⊕ y ⊕ −1). We claim LogK (Cf ) = T ∩ Q2 .
A point of Cf is of the form (p, −p − t) for some p ∈ K ∗ . If − val(p) 6= −1
then val(−p−t) = min{val(p), 1}, so (− val(p), − val(−p−t)) = (− val(p), −1)
28
if − val(p) < −1 and (− val(p), − val(−p−t)) = (− val(p), − val(p)) if − val(p) >
−1. In both cases (− val(p), − val(−p − t)) lies in T . If − val(p) = −1 we
have − val(−p − t) ≤ − min{val(p), 1} = −1, so (− val(p), − val(−p − t)) lies
in T . As val(K) = Q we get LogK (Cf ) ⊂ T ∩ Q2 .
We have
T ∩ Q2 = {(−q, −1)|q ∈ Q≥1 } ∪ {(−1, −q)|q ∈ Q≥1 } ∪ {(q, q)|q ∈ Q≥−1 }.
For q ∈ Q<−1 we have (t−q , −t−q −t) ∈ Cf and LogK (t−q , −t−q −t) = (q, −1).
For q ∈ Q>−1 we have (t−q , −t−q − t) ∈ Cf and LogK (t−q , −t−q − t) = (q, q).
For q ∈ Q≤−1 we have (−t−q −t, t−q ) ∈ Cf and LogK (t−q , −t−q −t) = (−1, q).
Thus LogK (Cf ) = T ∩ Q2 .
3.2
Tropical polynomials
Now we look at the algebra of tropical polynomials extending section 1.3.
We have delayed this section because it is necessary to use the notion of a
tropical hypersurface.
Let T[x1 , ..., xn ] be the set of tropical polynomials in the variables x1 , ..., xn .
If f = ⊕i∈I ai xi11 · · · xinn and g = ⊕j∈J bj xj11 · · · xjnn we put
f ⊕ g = ⊕k∈K (ak ⊕ bk ) xk11 · · · xknn where K = I ∪ J and ak = −∞ if
k∈
/ I. We also put f g = ⊕k∈K ck xk11 · · · xknn where K = I + J and
ck = ⊕i,j;i+j=k ai bj .
Example 3.6. Let f = x ⊕ y ⊕ −1 and g = −x ⊕ −y ⊕ −1, then
f g = −1 −x ⊕ −1 −y ⊕ 0 ⊕ x −y ⊕ y −x ⊕ −1 x ⊕ −1 y.
The tropical curve associated to f g can be seen in figure 3.1. As expected
it is the union of the tropical curves of f and g.
The proof that a polynomial ring over a semiring is a semiring is identical
to the proof that a polynomial ring is a ring so we can note the following
fact.
Proposition 3.7. The set T[x1 , ..., xn ] is a commutative semiring for the
operations ⊕ and .
In order to use addition and multiplication of tropical polynomials we
need to look at the functions the define on Rn . In case all coefficients are
zero the statement about the union of the tropical hypersurfaces is essentially
proposition 7.12 in [26].
29
Figure 3.1: The tropical curve defined by (x ⊕ y ⊕ −1) (−x ⊕ −y ⊕ −1).
Proposition 3.8. If f and g are tropical polynomials and v ∈ Rn then
f ⊕ g(v) = max{f (v), g(v)} and f g(v) = f (v) + g(v). Furthermore
V (f g) = V (f ) ∪ V (g).
Proof. Pick v ∈ Rn then
max{f (v), g(v)} = max{max{ai + i · v}, max{aj + j · v}}
i∈I
j∈J
= max {max{ak , bk } + k · v} = (f ⊕ g)(v),
k∈I∪J
which proves the first assertion.
Suppose ak + k · v is a maximal term in f (v) and al + l · v is a maximal
term in g(v). Thus we have ai + i · v + bj + j · v ≤ ak + k · v + bl + l · v
for all i ∈ I and j ∈ J. If i + j = k + j this gives ai + bj ≤ ak + bl so the
coefficient ck+j = maxi,j:i+j=k+j {ai + bj } of f g at k + j is equal to ak + bl .
From the same inequality it follows that ak + bl + (k + l) · v is a maximal
term of f g(v). This proves the second assertion.
If v ∈ V (f ) then there are distinct k, k 0 ∈ I such that ak + k · v and
ak0 + k 0 · v are maximal. If bl + l · v is a maximal term of g at v then
ak + bl + (k + l)v and ak0 + bl + (k 0 + l)v are maximal terms of f g at v so
v ∈ V (f g). The case v ∈ V (g) is identical. If v ∈
/ V (f ) ∪ V (g) then f and
g are linear on a neighborhood of v so f ⊕ g is as well, so we have proved
the third assertion.
30
The following fact will be used together with lemma 3.2.
Lemma 3.9. Let f and g be tropical polynomials and let ck = maxi,j;i+j=k {ai +
bj } be the coefficient of f g at xk . If ck + k · x is maximal at a point
v∈
/ V (f g) then the maximum in ck = maxi,j;i+j=k {ai + bj } is unique.
Proof. By lemma 3.8 we have v ∈
/ V (f ) and v ∈
/ V (g) so we can pick i ∈ I
0
and j ∈ J such that ai0 + i · x < ai + i · x for every i0 ∈ I distinct from i and
aj 0 + j 0 · x < aj + j · x for every j 0 ∈ J distinct from j. As in the proof of
lemma 3.8 it follows that i + j = k and ck = ai + bj .
If ck = ai0 + bj 0 for i0 ∈ I − {i} and j 0 ∈ J − {j} then
ck + k · v = ai0 + bj 0 + (i0 + j 0 ) · v < ai + bj + (i + j) · v = ck + k · v,
so ck is indeed unique.
Definition 3.10. If K is a field with a valuation, f =
a Laurent polynomial then
M
trop(f ) =
− val(ai ) xi .
P
i∈I
ai zi with I ⊂ Zn
i∈I
It is necessary to use − val because if ai = 0 then − val(ai ) = −∞, so the
term − val(ai ) xi does not contribute to trop(f ) either.
The following example shows that we have to look at functions on Rn
rather than tropical polynomials.
Example 3.11. Let f = z + t and g = z + (−t + t2 ), then trop(f g) =
x2 ⊕ −2 x ⊕ −2 and trop(f ) trop(g) = x2 ⊕ −1 x ⊕ −2. However both
trop(f g) and trop(f ) trop(g) define the function max{−2, 2x} on R2 .
The following proposition explains why tropicalization is a meaningful
construction.
Proposition 3.12. If f and g are polynomials over a field with a valuation
then trop(f g) defines the same function on Rn as trop(f ) trop(g). In
particular we have V (trop(f g)) = V (trop(f )) ∪ V (trop(g)).
P
P
i
Proof.
Let
f
=
a
z
and
g
=
bj zi with I, J ⊂ Zn , then f g =
i
i∈I
i∈JP
P
c˜ zk with K = I + J and c˜k =
. Thus trop(f g) =
i,j;i+j=k ai bj L
Lk∈K k
k
k
k∈K − val(c˜k ) x . We also have trop(f ) trop(g) =
k∈K ck x , where
ck = max {− val(ai ) − val(bj )} = max {− val(ai bj )}.
i,j;i+j=k
i,j;i+j=k
31
By lemma 3.2 we get
− val(c˜k ) ≤ − min {val(ai bj )} = max {− val(ai bj )} = ck .
i,j;i+j=k
i,j;i+j=k
Thus trop(f g)(v) ≤ trop(f ) trop(g)(v) for every v ∈ Rn .
Let v ∈
/ V (trop(f ) trop(g)) and let ck + k · v be the maximal term of
trop(f ) trop(g) at v then the maximum in ck = maxi,j;i+j=k {− val(ai ) −
val(bj ) is unique by lemma 3.9. Thus by lemma 3.2 we have − val(c˜k ) = ck ,
so trop(f g)(v) ≥ − val(c˜k ) + k · v = trop(f ) trop(g)(v).
This shows trop(f g) and trop(f )trop(g) are equal on the complement of
V (trop(f ) trop(g)). As both functions are continuous they must be equal.
This proves the first assertion and the second one follows directly.
The same type of statement is false for the tropical sum of polynomials.
Example 3.13. Let f = z + t + t2 and g = z − t then trop(f ) ⊕ trop(g) =
(x⊕−1)⊕(x⊕−1) = (x⊕−1) while trop(f +g) = x⊕−2, so V (trop(f +g)) =
{−2} and V (trop(f ) ⊕ trop(g)) = {−1}.
3.3
Kapranov’s theorem
Now we can sharpen Puiseux’s theorem in two steps. The second step will
be Kapranov’s theorem. For a polynomial f over the field of Puiseux series
it gives a complete description of the amoeba Af in terms of a tropical
hypersurface.
The first step is basically Kapranov’s theorem in one dimension. It is an
easy consequence of Puiseux’s theorem and lemma 3.12.
Lemma 3.14. Let f ∈ C{{t}}[z] be a polynomial in one variable over the
field of Puiseux series and suppose v ∈ V (trop(f )) then f has a root u such
that − val(u) = v.
Proof. By lemma 3.12 we can assume f is monic. We prove the lemma by
induction on deg(f ). If deg(f ) = 1 then f = z−u and trop(f ) = x⊕− val(u),
so the result is clear, even in the case u = 0.
Suppose the lemma holds for polynomials of degree at most n. Let
deg(f ) = n + 1 and let v ∈ V (trop(f )). By Puiseux theorem we have a
root w of f . If − val(w) = v we are done. Otherwise we write f = (z − w)f 0 .
By the case n = 1 we have V (z − w) = {− val(w)} and by lemma 3.12 it
follows that V (trop(f )) = V (trop(f 0 )) ∪ {− val(w)}. Thus v ∈ V (trop(f 0 ))
so by induction we get a root u of f 0 such that − val(u) = v. As u is also a
root of f we are done.
32
For products of linear polynomials we could repeat the proof of lemma 3.14
with a suitable version of example 3.5, but this does not work for irreducible
polynomials.
The solution from [24] is to use an inclusion of K ∗ in (K ∗ )n and apply
lemma 3.14 to get a root of the pullback of the polynomial.
However the inclusion used in [24] does not always work because tropicalization does not always behave nicely with respect to pullbacks as we can see
in example 3.16. This can be fixed by requiring an additional assumption on
the inclusion.
P
Theorem 3.15 (Kapranov). If f = i∈I ai zi is a polynomial over the field
of Puiseux series in n variables then Af = V (trop(f )) ∩ Qn .
P
Proof. Let f = i∈I ai zi with I ⊂ Zn .
For p ∈ Cf let v = LogK (p) = (− val(p1 ), ..., − val(pn )) then − val(ai pi ) =
− val(ai ) + i · v for all i ∈ I. We have
trop(f )(v) = max{− val(ai ) + i · v}.
i∈I
(3.3.1)
and this maximum is attained for only one index in I if and only if the
k
minimum min
i p )} is attained by one index in I. If that is the case
P i∈I {val(a
we get val( i∈I ai pi )P
= mini∈I {val(ai pk )} by lemma 3.2. This is impossible
as ∞ = val(0) = val( i∈I ai pi ) and val(ai pk ) < ∞ for some i ∈ I. Thus we
have Af ⊆ V (trop(f )) ∩ Qn .
Now let v ∈ V (trop(f )) ∩ Qn . We construct an inclusion of K ∗ into (K ∗ )n
to apply lemma 3.14.
For any nonzero α ∈ Zn with coprime coefficients the function ϕ(s) =
−v1 α1
(t s , ..., t−vn sαn ) defines an inclusion ϕ : K ∗ → (K ∗ )n . If we define ψ : R →
Rn by ψ(λ) = v + λα then the diagram
K∗
− val
ϕ
/
(K ∗ )n
LogK
/
R
ψ
(3.3.2)
Rn
is commutative and ψ(0) = v. Thus it suffices to find an α such that
trop(ϕ∗ f ) has a root at 0.
To ensure this we pick an α ∈ Zn such that α · i 6= α · j for every pair
of distinct i, j ∈ I. As each of these equations defines the complement of
hypersurface it is possible to pick such an α. Now the pullback of f (z)
−vn αn
∗
is the polynomial g = ϕ∗ f given by g(z) = f (t−v1 z α1 , ...,
Pt z )k on K .
To get the tropicalization of g we expand it so g(z) =
k∈K bk z , where
33
P
bk = i∈I:α·i=k ai t−i·v . By the choice of α each sum bk has only one term so
− val(bk ) = − val(ai ) + i · v if there is a i ∈ I with α · i = k and bk = −∞
otherwise.
The restriction of trop(f ) to the image of ψ is given by trop(f )(v + λα) =
maxi∈I {− val(ai ) + i · (v + λα)} = maxi∈I {− val(ai ) + i · v + (i · α)λ}.
Thus trop(g) is equal to the restriction of trop(f ) and trop(g) is not locally
linear at 0, so g has a zero s with − val(s) = 0. Now ϕ(s) is a root of f
because g is the pullback of f and LogK (ϕ(s)) = v because diagram 3.3.2
commutes.
The following example shows that tropicalization does not always commute with pullbacks.
Example 3.16. Let f = tuw2 +(−t+t2 )u2 w+tu5 , then trop(f ) = max{−1+
x + 2y, −1 + 2x + y, −1 + 5x} and V (f ) is the curve in figure 3.2
Figure 3.2: The tropical curve defined by max{−1 + x + 2y, −1 + 2x + y, −1 +
5x}.
Pick α = (1, 1) and define ϕ and ψ as in diagram 3.3.2. Now α does not
satisfy the conditions required in the proof of Kapranov’s theorem.
The pullback ϕ∗ f of f to K ∗ is given by ϕ∗ f (z) = t2 z 3 +tz 5 , so trop(ϕ∗ f ) =
max{−2 + 3x, −1 + 5x} and the restriction of trop(f ) to R is given by
max{−1 + 3x, −1 + 5x}.
Thus trop(ϕ∗ f ) is locally linear at 0 and the root is at x = − 12 . Thus with
lemma 3.14 we get a root p of f with LogK (p) = (− 12 , − 12 ). In figure 3.2 we
can see that the root has shifted along one of the branches of V (trop(f )).
34
Even if trop commutes with the pullback lemma 3.14 can still give the
wrong root if the inclusion of R in Rn does not intersect V (f ) transversally.
Example 3.17. Let f = u + w + t, then trop(f ) = max{x, y, −1} and
v = (0, 0) ∈ V (trop(f )). Let α = (1, 1) and define ϕ and ψ as in diagram
3.3.2. Now ϕ∗ f (z) = 2z + t, so trop(ϕ∗ f ) = max{λ, −1}. Also trop(f ) =
max{x, y, −1}, so the restriction to R is given by max{λ, λ, −1} which is
equal to trop(ϕ∗ f ).
The tropical root of trop(ϕ∗ f ) is λ = −1, so the root p of f given by lemma
3.14 satisfies LogK (p) = (−1, −1), so it is not the root we were looking for.
Remark. Kapranov’s theorem holds for any algebraically closed field with
a non-Archimedean absolute value. The proof given here also works in the
general case. We proved 3.12 for arbitrary fields with a valuation, so the proof
of lemma 3.14 is identical in the general case. In the proof of Kapranov’s
theorem the elements t−v1 , ..., t−vn have to be replaced with elements of the
right absolute value. This is possible in an algebraically closed field.
35
Chapter 4
A limit of amoebas
In this chapter we study the the limit of a collection of amoebas. We follow
an article [14] by Grigory Mikhalkin. This approach makes it possible to
construct algebraic curves which have certain topological properties.
This kind of construction was first used by Oleg Viro to construct real
algebraic curves with a particular topology in [19]. In [14] it is used to find
the topology of an arbitrary smooth hypersurface in (K ∗ )n .
4.1
The construction of the limit
P
j
n
Let f =
j∈Q aj z be a Laurent polynomial, where Q ⊂ Z is the set
{j ∈ Zn |aj 6= 0}. We make a limit of amoebas by changing both the base
of the logarithm and the coefficients of f . Let Cf be the curve defined by f
and let Af be the amoeba of f . For r ∈ R>1 we define Logr : Cn → Rn by
Logr (z1 , ..., zn ) = (logr |z1 |, ..., logr |zn |), then Logr (Cf ) = Af ·M , where M is
1
. Thus Logr (Cf ) is homeomorphic with
the diagonal matrix with entries log(r)
P
Af . The coefficients of f are changed by the formula fr = j∈Q aj rv(j) zj for
some function v : Q → Q. If Xr is the hypersurface defined by fr , then we
define Ar = Logr (Xr ).
P
−v(j) j
We can also look at the tropical polynomial ft (z) =
z ∈
j∈Q aj t
±
C{{t}}[z ]. The fact that the minus sign appears in the exponent is an
unfortunate consequence of our conventions in the Log map and the definition
of the tropicalization of a polynomial.
The polynomial ft is called the patchworking polynomial. As C{{t}}
is a field with a non-archimedean norm, the polynomial ft defines a nonarchimedean amoeba. By Kapranov’s theorem the closure of this amoeba is
the tropical curve defined by trop(ft ) = maxj∈Q {v(j) + j · x}, where x =
(x1 , ..., xn ) is a real variable. We denote this tropical curve with AK .
36
The goal of this chapter is to prove that for r → ∞ the amoebas Ar
converge to AK .
Example 4.1. We can already compute an easy example. Let f = u + w + 1
and v(0, 0) = v(1, 0) = v(0, 1) = 0, then Ar = M · Af , where M is the
1
. It is easy to see these set converge to the
diagonal matrix with entries log(r)
union of the three lines given by {((−t, 0)|t ∈ R≥0 )} ∪ {((0, t)|t ∈ R≥0 )} ∪
{((−t, −t)|t ∈ R≥0 )}, which is the tropical curve given by x⊕y ⊕0 = trop(f ).
Figure 4.1: The set Ar with r = e7 in blue and the tropical curve of x ⊕ y ⊕ 1
in red.
In case f is a Laurent polynomial in two variables we have vol(Af ) ≤
π2
π vol(Newt(f )) by [18]. Thus we get vol(Logr (Cf )) ≤ log(r)
n vol(Newt(f )),
so the volume of Logr (Cf ) tends to zero when r goes to ∞. This also happens
when we change the coefficients of f as the newton polytope remains the
same. Thus we know that if limr→∞ Ar exists it has zero volume.
Furthermore proposition 2.6 and theorem 1.28 together with the preceding
example already suggest a strong relation between the limit and a tropical
curve.
2
37
4.2
The Hausdorff distance
We need to make precise what it means for a sequence of sets to converge,
this is expressed by the Hausdorff distance.
Definition 4.2. The Hausdorff distance between two closed sets A, B ⊂ Rn
is given by
max{sup(d(a, B)), sup(d(b, A))}.
a∈A
b∈B
The Hausdorff distance does not give a metric on the set of closed subsets
of Rn because it is not always finite. However it does satisfy the triangle
inequality. For a collection {Ar }r∈R>1 we can define the limit as the closed
set B ⊂ Rn such that limr→∞ dH (Ar , B) = 0 if such a set exists. This set is
unique by the following lemma.
Lemma 4.3. If {Ar }r∈R>1 converges to B then B consists of all b ∈ Rn for
which there are ar ∈ Ar such that limr→∞ ar = b.
Proof. By definition of dH for every b ∈ B we can find a ar ∈ Ar with
d(ar , b) ≤ dH (Ar , B), so limr→∞ ar = b.
If b ∈
/ B then d(b, B) > ε for some ε ∈ R>0 . By the convergence of the
Ar we have an M ∈ R>0 such that dH (Ar , B) ≤ 2ε for all r > M . Thus also
supa∈ Ar d(a, B) ≤ 2ε . By the triangle inequality we get that d(b, Ar ) > 2ε for
r > M , so b is not the limit of points in the sets Ar .
The converse of the lemma does not hold, for example if Ar = [0, r] then
the set of limit points is [0, ∞), but dH ([0, r], [0, ∞)) = ∞ for every r ∈ R>0 .
4.3
A neighborhood of the tropical curve
Now we can construct a collection of decreasing neighborhoods of AK which
contain Ar . Let M ∈ R>1 and let ArK be the subset of Rn described by the
inequalities
ck + k · x ≤ max {cj + j · x} + logr (M ),
j∈Q−{k}
where k ranges over Q and cj = v(j) + logr |aj |. To ensure both AK ⊂ ArK
and Ar ⊂ ArK we pick M = max{N, Mf2 }, where N = #Q − 1 and Mf =
maxj∈Q {|aj |, |aj |−1 }.
Lemma 4.4. The set ArK is a neighborhood of AK for every r ∈ R>1 and
limr→∞ dH (ArK , AK ) = 0.
38
Proof. The tropical curve AK is given by the inequalities
v(k) + k · x ≤ max {v(j) + j · x},
j∈Q−{k}
so if y ∈ AK we get ck + k · y ≤ maxj∈Q−{k} {v(j) + j · y} + logr |ak |. We
have logr |ak | ≤ logr (Mf ) and maxj∈Q−{k} {v(j) + j · y} ≤ maxj∈Q−{k} {v(j) +
logr |aj | + j · y} + logr (Mf ). This shows AK ⊂ ArK for every r ∈ R>0 and
this also implies supy∈AK (d(y, ArK )) = 0.
Now we estimate d(y, AK ) for y ∈ ArK . If y ∈ AK , then d(y, AK ) = 0.
Thus we can assume y is a the component of the complement of AK defined
by v(k) + k · x > maxj∈Q−{k} {v(j) + j · x} for some k ∈ Q. Pick j1 ∈ Q − {k}
such that v(j1 ) + j1 · y = maxj∈Q−{k} {v(j) + j · y}.
By definition of ArK we have v(k) + logr |ak | + k · y ≤ maxj∈Q−{k} {v(j) +
logr |aj | + j · y} + logr (M ), so v(k) + k · y ≤ v(j1 ) + j1 · y + 3 logr (M ). Now let
logr (M )
0
0
0
y0 = y− 3||k−j
2 (k−j1 ). We have v(k)+k·y ≤ v(j1 )+j1 ·y , so y does not lie
1 ||
in the same component of AcK as y. Thus d(y, AK ) ≤ d(y, y0 ) = 3 logr (M ).
This shows supy∈ArK d(y, AK ) ≤ 3 logr (M ), which proves the lemma.
We only need the sets ArK to show the convergence of the Ar and the
following two properties are used in the proof of theorem 4.7.
Lemma 4.5. For every r ∈ R>0 we have Ar ⊂ ArK .
Proof. If P
x ∈ Ar we can pick z ∈ (C∗ )n with P
fr (z) = 0 and Logr (z) = x.
v(j) j
v(k) k
We have j∈Q aj r z = 0, so |ak r z | = | j∈Q−{k} aj rv(j) zj | for every
k ∈ Q. Applying logr and the triangle inequality gives
X
ck + k · Logr (z) = logr (|
aj rv(j) zj |)
j∈Q−{k}
≤ logr (
X
|aj rv(j) zj |)
j∈Q−{k}
= logr (
X
|rcj ||zj |)
j∈Q−{k}
≤ logr (N max |rcj ||zj |)
j∈Q−{k}
=
max {cj + j · Logr (z)} + logr (N ),
j∈Q−{k}
so x = Logr (z) ∈ ArK .
Lemma 4.6. If the components of the complement of ArK given by cki +ki ·x >
maxj∈Q−{ki } {cj + j · x} for k1 , k2 ∈ Q and k1 6= k2 are nonempty they lie in
distinct components of the complement of Ar .
39
Proof. Let x be in the component given by cki + ki · x > maxj∈Q−{ki } {cj +
j · x} + logr (M ) an let Logr (z) = x. We have rcj +j·x
= rv(j) |aj ||zj |, so we
P
get rv(ki ) |aki ||zki | > M maxj∈Q−{ki } {rv(j) |aj ||zj |} > | j∈Q−{ki } rv(j) aj zj | by
the choice of M . Now we can apply theorem 2.9, so x lies in the component
of Ar corresponding to ki . By the same theorem the components of Ar
corresponding to k1 and k2 are distinct which proves the lemma.
4.4
The limit
Now we can prove the statement of the introduction and construct some
curves with nice amoebas.
Theorem 4.7. When r → ∞ the amoebas Ar converge to AK for the Hausdorff distance.
Proof. By lemma 4.4 and 4.5 we have limt→∞ supy∈Ar d(y, AK ) = 0.
Let ε > 0. Because AK is a finite collection of subsets of hypersurfaces
and limr→∞ dH (ArK , AK ) = 0 we can find a T > 0 such that ArK does not
contain any ball of diameter ε.
Now let x ∈ AK . Because the components of the complement of ArK
are convex we can find a ball B of diameter ε which intersects two distinct
components C1 and C2 of the complement of ArK . If B contains no point of
Ar for some r > T we get that C1 and C2 are connected in the complement
of Ar . This is impossible by lemma 4.6. Thus d(x, Ar ) < ε.
We now consider an example.
Example 4.8. Consider the curve defined by f (u, w) = u3 + w3 + λuw + 1.
If λ = 0 then Af is the image of the amoeba u + w + 1 under a linear map
by corollary 5.2 and therefore the complement has no bounded components.
In figure 4.2 we can see that the amoeba also has no holes for small values of
λ. 1 Nu we pick the function v by v(0, 0) = v(3, 0) = v(0, 3) and v(1, 1) = 1
and note that Ar is the amoeba of u3 + w3 + ruw + 1 multiplied coordinate
1
. Now theorem 4.7 shows that Acf has a bounded component
wise with log(r)
for sufficiently large values of λ. The case of λ = 2 is figure 2.1.
We can also use theorem 4.7 to show that the maximum number of components given by corollary 2.4 is achieved by some curves.
Theorem 4.9. For every finite set Q ⊂ Zn there is a Laurent polynomial
g with Newt(g) = conv(Q) such that the number of components of Ag is
# Newt(g) ∩ Zn .
1
This could in principle be proved by looking at the critical points of Log map.
40
Figure 4.2: The amoeba of u3 + w3 + λuw + 1 for λ = 1.
P
Proof. Let I = Newt(g) ∩ Zn and let f ((z)) = i∈I zi . By proposition 1.11
there is a function v : I → Z such that the subdivision of Newt(f ) = Newt(g)
induced by the tropical curve V given by maxi∈I {ai + i · x} has I as set of
vertices. By theorem 1.28 the components of the complement of V correspond
bijectively to I. Now we apply theorem 4.7 and note AK = V .
Thus for sufficiently large
that Acr has at least #I components.
P r itv(i)follows
The amoeba of g((z)) = i∈I r zi is homeomorphic to Acr so it follows that
Acg has exactly #I components.
Now we give an example of an amoeba constructed this way.
Example 4.10. We want to construct a polynomial with Newton polytope
N = conv{(0, 0), (3, 0), (0, 3), (2, 3),
maximal number of compoP(3, 2)}j1 and
2
j2
nents. Let Q = N ∩Z and let f = j∈Q u w . Now we must pick a function
v : Q → Q such that trop(ft ) has a maximal number of components. If we
pick v as in example example 1.12 then trop(ft ) will be the tropical polynomial from example 1.20 which indeed has the maximum possible number of
components.
The convergence of the modified amoebas Ar to the tropical curve can be
seen in figure 4.3.
For sufficiently large r the complement of Ar has a component for every
component of the complement of the tropical curve of ft by lemma 4.6.
As Ar is the amoeba of fr multiplied with a constant we can use any fr
41
(a) r=2
(b) r=3
(c) r=5
(d) r=100
Figure 4.3: Converging amoebas.
42
with sufficiently large r, for example f14 .
f14 = u3 + w3 + 1 + u2 w3 + u3 w2 + 14u + 14u2 + 14w + 14w2 + 14u3 w
+14uw3 + 196uw + 196u2 w + 196uw2 + 196u2 w2
Figure 4.4: The amoeba of f14 .
43
Chapter 5
Appendix I
In this appendix we consider the effect of an endomorphism of (C∗ )2 on the
amoeba of a curve. This is the special case n = d = 2 of section 4 of [25],
although the formulation is slightly different.
We are interested in the endomorphisms of (C∗ )2 with the structure of
a maximal ideal spectrum and not of an algebraic group, so multiplication
with a constant is an endomorphism.
5.1
Endomorphisms of (C∗)2
The points of (C∗ )2 correspond to the closed points of spec(C[u, w, u−1 , w−1 ]),
which are by definition the maximal ideals of C[u, w, u−1 , w−1 ]. By Hilbert’s
nullstellensatz and elementary properties of localization the set of maximal
ideals of C[u, w, u−1 , w−1 ] is {(u − λ1 , w − λ2 )|λ1 , λ2 ∈ C∗ }. The correspondence between maximal ideals and points is given by mapping a maximal
ideal m to the single element of {(λ1 , λ2 ) ∈ (C∗ )2 |f (λ1 , λ2 ) = 0 for all f ∈ m},
thus (u − λ1 , w − λ2 ) 7→ (λ1 , λ2 ).
For M ∈ Mat2 (Z) and α = (α1 , α2 ) ∈ (C∗ )2 we define
(α1 , α2 )M = (α1m11 α2m21 , α1m12 α2m22 ).
By definition the endomorphisms of (C∗ )2 correspond the endomorphisms
of C[u, w, u−1 , w−1 ] as C-algebra and we can use commutative algebra to
describe these.
Lemma 5.1. The set End(C[u, w, u−1 , w−1 ]) consists of all maps defined by
f (z) 7→ f (α · zM ),
with M ∈ Mat2 (Z) and α = (α1 , α2 ) ∈ (C∗ )2 , where f (z) = f (u, w) ∈
C[u, w, u−1 , w−1 ]. Furthermore all these maps are distinct.
44
Proof. Denote A = C[u, w, u−1 , w−1 ]. By the universal property of localization, homomorphisms ϕ : A → A are the unique extensions of homomorphisms ϕ̃ : C[u, w] → A such that ϕ̃(u) and ϕ̃(w) are units. The set of units
is {αij ui wj |αij ∈ C∗ and i, j ∈ Z}. Thus a homomorphism ϕ̃ : C[u, w] → A
extends to an endomorphism of A if and only if ϕ̃(u) = α1 um11 wm21 and
ϕ̃(w) = α2 um12 wm22 with α1 , α2 ∈ C∗ and m11 , m21 , m12 , m22 ∈ Z. In this
case ϕ̃ and its extension ϕ : A → A are given by f (z) 7→ f (α · zM ). This
proves the first statement in the lemma.
If ϕ is the endomorphism given by M ∈ Mat2 (Z) and α = (α1 , α2 ) ∈
(C∗ )2 then ϕ(u) = α1 um11 wm21 and ϕ(w) = α2 um12 wm22 , so M and α are
unique.
We can use this lemma to show the group of automorphisms is given by a
semidirect product. It is easy to check (αβ)M = αM β M and αM N = (αN )M
for all α, β ∈ (C∗ )2 and M, N ∈ Mat2 (Z). This way we get a homomorphism
GL(2, Z) → AutGrp ((C∗ )2 ) and thus a semidirect product (C∗ )2 o GL(2, Z)
of groups.
Proposition 5.2. The map Ψ : (C∗ )2 o GL(2, Z) → Aut(C[u, w, u−1 , w−1 ]),
defined for (α, M ) ∈ (C∗ )2 o GL(2, Z) and f (z) = f (u, w) ∈ C[u, w, u−1 , w−1 ]
by
Ψ(α, M )(f (z)) = f (α · zM ),
(5.1.1)
is an isomorphism.
Proof. By the previous lemma we know that formula 5.1.1 defines an endomorphism of C[u, w, u−1 , w−1 ]. It is clear that Ψ maps the unit element of
(C∗ )2 o GL(2, Z) to the identity map, so if Ψ is multiplicative all endomorphisms defined by formula 5.1.1 are automorphisms.
Let ψ and ϕ be the endomorphisms defined by (β, N ) and (α, M ) with
β, α ∈ (C∗ )2 and N, M ∈ Mat2 (Z) then we have
ψ ◦ ϕ(u) = α1 β1m11 β2m21 um11 n11 +m21 n12 wm11 n21 +m21 n22
(5.1.2)
ψ ◦ ϕ(w) = α2 β1m12 β2m22 um12 n11 +m22 n12 wm12 n21 +m22 n22 .
(5.1.3)
and
This shows ψ ◦ ϕ is the endomorphism defined by (αβ M , N M ). In particular
we have Ψ(β, N )◦Ψ(α, M ) = Ψ(αβ M , N M ) = Ψ((β, N )(α, M )), which shows
that Ψ is well defined and a homomorphism.
If ϕ is an endomorphism and ψ = ϕ−1 it follows by equation 5.1.2 and
5.1.3 that N M is the unit matrix, so N ∈ GL(2, Z) and ϕ ∈ Im(Ψ). Finally
Ψ is surjective by the uniqueness of (α, M ) in the previous lemma.
45
Equations 5.1.2 and 5.1.3 also show that the composition of endomorphisms is also well behaved with respect to matrix multiplication, so would
also be possible to give a description of End(C[u, w, u−1 , w−1 ]) in terms of
the semigroups (C∗ )2 and Mat2 (Z).
For every endomorphism of C[u, w, u−1 , w−1 ] we have an endomorphism
of the maximal spectrum. The following proposition shows how this map behaves with respect to the identification of the maximal spectrum with (C∗ )2 .
Proposition 5.3. If ϕ is the endomorphism of C[u, w, u−1 , w−1 ] defined by
α ∈ (C∗ )2 and M ∈ Mat2 (Z) then the corresponding endomorphism ϕ∗ of
(C∗ )2 is given by ϕ∗ (λ) = αλM .
Proof. Denote A = C[u, w, u−1 w−1 ] and let msp(A) be the maximal ideal
spectrum of A. If ϕ in an endomorphism of A the canonical map ϕ̃ : msp(A) →
msp(A) is given by ϕ̃(m) = ϕ−1 (m) for maximal ideals m of A.
We have identified (C∗ )2 with msp(A) by the map Z : msp(A) → (C∗ )2
which maps a maximal ideal to its set of zeros. Thus we need to show that
the diagram
Z
/ (C∗ )2
msp(A)
ϕ̃
msp(A)
Z
/
ϕ∗
(C∗ )2
commutes. Let m ∈ msp(A), then m = (u − λ1 , w − λ2 ) for some λ1 , λ2 ∈ C∗ ,
so Z(m) = (λ1 , λ2 ) = λ and ϕ∗ Z(m) = αλM . If f ∈ ϕ−1 (m) then f (αλM ) =
ϕ(f )(λ), so αλM is the zero of ϕ−1 (m).
Proposition 5.4. Let ϕ be the endomorphism of C[u, w, u−1 , w−1 ] defined
by α ∈ (C∗ )2 and M ∈ Mat2 (Z). If det(M ) 6= 0 then the corresponding
endomorphism ϕ∗ of (C∗ )2 is surjective.
Proof. It is enough to to prove this with α = (1, 1). By the Smith normal
form theorem we can find A, B, ∈ GL(2, Z) and D ∈ Mat2 (Z) such that
AM B = D and D is a diagonal matrix. From the assumptions it follows
directly that det(D) 6= 0, so D defines a surjective endomorphism of (C∗ )2 .
The equation AM B = D also holds when we consider the maps defined
by A, M, B, D, so M is also surjective.
The above argument can easily be turned into an equivalence.
Proposition 5.5. Let ϕ∗ be a surjective endomorphism of (C∗ )2 as in the
previous proposition. If Cf ⊂ (C∗ )2 is the curve defined by f ∈ C[u, w, u−1 , w−1 ]
then the restriction ϕ∗ |Cϕ(f ) : Cϕ(f ) → Cf is surjective.
Proof. We have f (ϕ∗ (λ)) = ϕ(f )(λ) for every λ ∈ (C∗ )2 , so λ ∈ Cϕ(f ) if and
only if ϕ∗ (λ) lies in Cf . The corollary follows because ϕ∗ is surjective.
46
5.2
The transformation of the amoeba
It turns out that the surjective maps of curves in the previous section give
linear bijections between the amoebas of the curves.
First we give the action on the Newton polytope.
Proposition 5.6. Let ϕ is the endomorphism of C[u, w, u−1 , w−1 ] defined by
α ∈ (C∗ )2 and M ∈ Mat2 (Z), then Newt(ϕ(f )) = M Newt(f )
Proof. The monomial ui wj occurs in f if and only if the monomial ϕ(ui wj ) =
un11 i+n12 j wn21 i+n22 j occurs in ϕ(f ). As a linear map maps the convex hull of
a set to the convex hull of the image of the set, the proposition follows.
The map ϕ∗ |Cϕ(f ) : Cϕ(f ) → Cf from corollary 5.5 translates directly to a
map between the amoebas of f and ϕ(f ). Because ϕ∗ is surjective this map
becomes a bijection.
Corollary. Let Cf ⊂ (C∗ )2 be the curve defined by f ∈ C[u, w, u−1 , w−1 ]
and let ϕ be the endomorphism of C[u, w, u−1 , w−1 ] defined by α ∈ (C∗ )2 and
M ∈ Mat2 (Z) with det(M ) 6= 0, then
Af = M Aϕ(f ) − Log(α).
Proof. By proposition 5.5 the map ϕ∗ |Cϕ(f ) : Cϕ(f ) → Cf is surjective and
∗
byproposition
(λ) = αλM . Let T : R2 → R2 be defined by
5.3
wehave ϕ x
x
log |α1 |
T
=M·
−
, then it is easy to check that the diagram
y
y
log |α2 |
Cϕ(f )
ϕ∗
/ Cf
Log
R2
T
/
Log
R2
commutes, which shows the sets Log(Cf ) and T Log(Cϕ(f ) ) are equal.
Example 5.7. Corollary 5.2 makes it possible to drawsomeamoebas in two
3 5
different ways. Consider f = u + w + 1 and M =
, then ϕ(f ) =
7 2
u3 w7 + u5 w2 + 1.
In figure 5.1 the amoeba of f has been translated by M , however the
distribution of the computed points has become rather uneven as only a few
points of the left tentacle have been computed.
47
Figure 5.1: The amoeba of f translated by M .
Directly computing points of the amoeba of u3 w7 + u5 w2 + 1 is slower
because of the higher degree, but the points are distributed evenly as can be
seen in figure 5.2.
Figure 5.2: The amoeba of u3 w7 + u5 w2 + 1 computed directly.
The difference in scale is because both the amoeba of f and ϕ(f ) were
computed in the square defined by −4 ≤ x, y ≤ 4.
48
Chapter 6
Appendix II
In this appendix we give the Sage [23] code used to draw the amoebas in this
thesis. It is very similar to the matlab code available at [1].
From a practical point of view the use of Sage it convenient because it has
a number built in operations on polynomials. Furthermore it is open source
and free to use.
6.1
Computing the amoeba of a planar curve
The following code can be put directly into a worksheet, except for an extra
linebreak in line 9.
In the first box we declare the box [xmin, xmax] × [ymin, ymax] in which
we want to draw the amoeba, the variables, the number of points drawn and
the polynomial f .
1
2
3
4
5
6
7
8
9
10
var ( ’ xmin ’ , ’ xmax ’ , ’ ymin ’ , ’ ymax ’ )
xmin=−3
xmax=3
ymin=−3
ymax=3
var ( ’ a n g l e s t e p ’ , ’ g r i d s t e p ’ )
g r i d s t e p =250
a n g l e s t e p =120
var ( ’ u ’ , ’ w’ )
f =4∗u∗w−uˆ2∗w−wˆ2∗u−1
The following box computes points along vertical lines in the chosen box.
Because we only look at polynomials with real coefficients the angle of the
possible solutions can be taken in [0, π).
49
1 x g r i d s t e p s i z e =(xmax−xmin ) / g r i d s t e p
2 List x =[]
3 f o r j i n range ( 1 , g r i d s t e p ) :
4
r=xmin+j ∗ x g r i d s t e p s i z e
5
f o r k i n range ( 0 , a n g l e s t e p ) :
6
t h e t a =(k/ a n g l e s t e p )∗ p i ∗ I
7
z1=e ˆ r ∗ e ˆ ( t h e t a )
8
g=f . s u b s t i t u t e ( u=z1 )
9
S=g . r o o t s (w, r i n g=ComplexField ( p r e c =10)
10
, m u l t i p l i c i t i e s=F a l s e )
11
f o r p2 i n S :
12
l o g p 2=l o g ( abs ( p2 ) )
13
i f logp2>ymin and logp2<ymax :
14
L i s t x . append ( ( r , l o g p 2 ) )
We can now plot the list of points with the following command.
1
l i s t p l o t ( Lx , s i z e =2, a s p e c t r a t i o =1)
If we only compute points along vertical lines any tentacles of the amoeba
which are vertical will be drawn badly because the become thinner than the
space between the lines on which the points are computed. This can be seen
in the next example.
Example 6.1. If we run the algorithm for f = 1 + u3 + w3 + 2uw in the box
[−3, 3] × [−3, 3] with grid step = 250 and angle step = 120 we get figure 6.1
50
Figure 6.1: Incomplete amoeba for f = 1 + u3 + w3 + 2uw.
Thus we have to add a second list of points, now computed on horizontal
lines. Then we get the full picture. We can plot both list with the following
command.
1
l i s t p l o t ( L i s t x )+ l i s t p l o t ( L i s t y )
The complete amoeba of example 6.1 is figure 2.1
6.2
Critical points
If we compute too few point to fill up the amoeba we get a picture where we
can see some indication as to how the curve is folded onto the amoeba. For
example in figure 6.2 we can see that the boundary of the hole in the amoeba
is the union of three foldlines which continue in the amoeba.
51
Figure 6.2: Some points on the amoeba of f = 1 + u3 + w3 + 1.25uw.
These lines are critical points of the Log map on the curve. In [25] an
algorithm to compute these lines is given.
In the following amoeba the boundary of the hole in the amoeba seems to
be smooth.
52
Figure 6.3: Some points on the amoeba of f = 4uw − u2 w − w2 u − 1.
53
Populaire samenvatting
Een vlakke kromme is de verzameling van punten in het vlak die aan een
bepaald soort vergelijking voldoen. In figuur 6.4(a) staan bijvoorbeeld de
oplossingen van de vergelijking y − x2 − 2 = 0. In dit geval is de kromme
een parabool. 1 In het algemeen bestaat een kromme uit een aantal lijnen
en cirkels.
In figuur 6.4(b) staat een geschaalde versie2 van dezelfde kromme. Vanuit
dit perspectief lijkt de kromme uit twee rechte lijnen te bestaan.
(a) parabool
(b) geschaalde parabool
Figure 6.4
Om te zien met welke rechte lijnen we te maken hebben bekijken we een
tropische kromme. Deze naam heeft niets met een regenwoud te maken
maar met het feit dat dit soort meetkunde vernoemd is naar de Braziliaanse
wiskundige Imre Simon.
In figuur 6.5 is aangegeven in welk deel van het vlak de functies 2x, y en
log(2) maximaal zijn. De punten waar meerdere van deze functies maximaal
zijn vormen de drie rode lijnstukken. Dit is de tropische kromme van 2x, y
en log(2). We zien dat twee van de lijnen figuur 6.4(b) goed benaderen.
1
Om technische redenen kijken we alleen naar oplossingen die niet op de coördinaatassen
liggen.
2
We sturen een punt (x, y) naar (log |x|, log |y|).
54
Figure 6.5: Een tropisch kromme.
Nu gaan we uitzoeken waar de derde lijn vandaan komt.
In figuur 6.4(a) hebben we oplossingen van y−x2 −2 = 0 met reële getallen
bepaald en dit geeft een kromme in het vlak. We kunnen ook zoeken naar
oplossingen met complexe getallen. Als we dit doen geven de oplossingen
een oppervlak in een vierdimensionale ruimte, waar we dus geen plaatje van
kunnen tekeken.
Wel kunnen we weer dezelfde soort schaling toepassen als we afbeelding
6.4(b) gedaan hebben. De vierdimensionale ruimte wordt nu naar een tweedimensionale ruimte afgebeeld. Het beeld van verzameling van oplossingen
is nu te zien in figuur 6.6. We kunnen nu opmerken dat de lijn in figuur
6.4(b) de bovenrand van figuur 6.6 is en dat de we de tak gevonden hebben
die hoort bij de derde lijn van de tropische kromme.
55
Figure 6.6: De amoeba van de vergelijking y − x2 − 2 = 0.
De figuur die we krijgen als we de procedure van afbeelding 6.6 toepassen
op de complexe oplossingen van een vergelijking heet de amoeba van de
vergelijking. De gelijkenis met de eencellige organismes met dezelfde naam
wordt duidelijk als we naar een ander voorbeeld van een amoeba kijken,
bijvoorbeeld afbeelding 6.7.
Figure 6.7: De amoeba van de vergelijking x3 + y 3 + x2 y 4 + x4 y 3 + 3.8x3 y +
12.9x2 y 2 + 4.55xy 3 + 4.6x3 y 3 = 0.
56
In deze sciptie hebben we een constructie van Grigory Mikhalkin bekeken
waar een amoeba steeds verder vervormd wordt tot die steeds meer op een
tropische kromme gaat lijken. Het effect van deze constructie is duidelijk te
zien in afbeelding 6.8.
(a) r=2
(b) r=3
(c) r=5
(d) r=100
Figure 6.8: Het vervormen van een amoeba.
Ook hebben we het geval bekeken waar we niet kijken naar oplossingen
van vergelijkingen met complexe getallen, maar met oplossingen in een ander
getalsyteem. Dit geval wordt geheel beschreven door de stelling van Kapranov.
57
Bibliography
[1] O. Aleksandrov, matlab code, available
wikimedia.org/wiki/File:Amoeba3.png.
at
http://commons.
[2] G. Bergman, The logarithmic limit-set of an algebraic variety,
Transactions of the American Mathematical Society 157 (1971),
available at http://www.ams.org/journals/tran/1971-157-00/
S0002-9947-1971-0280489-8/.
[3] F. Cools, J. Draisma, S. Payne, E. Robeva, A tropical proof of the BrillNoether Theorem, Advances in Mathematics 230 (2012), available at
http://arxiv.org/abs/1001.2774.
[4] M. Einsiedler, M. M. Kapranov, and D. Lind, Non-Archimedean amoebas and tropical varieties, J. Reine Angew. Math. 601 (2006), available
at http://arxiv.org/abs/math/0408311.
[5] L. Dale, Monic and monic free ideals in a polynomial semiring,
Proceedings of the American Mathematical Society 56 (1976),
available at http://www.ams.org/journals/proc/1976-056-01/
S0002-9939-1976-0404354-8/S0002-9939-1976-0404354-8.pdf.
[6] D. Eisenbud, Commutative algebra with a view towards algebraic geometry, Graduate Texts in Mathematics 150, Springer-Verlag, Berlin,
1995.
[7] M. Fosberg, M. Passare, A. Tsikh, Laurent determinants and arrangements of hyperplane amoebas, Advances in Mathematics 151
(2000), 45–70, available at http://www.sciencedirect.com/science/
article/pii/S000187089991856X.
[8] I.M. Gelfand, M.M. Kapranov, A.V. Zelevinsky, Discriminants, resultants, and multidimensional determinants, Birkhäuser, 1994.
[9] Z. Izhakian, L. Rowen, Ideals of polynmial semirings in tropical algebra.,
preprint, available at http://arxiv.org/abs/1111.6253.
58
[10] M.M. Kapranov, Amoebas over non-Archimedean fields, unpublished.
[11] J. Korevaar, J. Wiegerinck, Several complex variable, manuscript, version of setptember 5, 2013, available at http://staff.science.uva.
nl/~janwieg/edu/scv/.
[12] J.A. de Loera, Nonregular triangulations of products of simplices,
Discrete & Computational Geometry volume 15 3 (1996), available
at
https://www.math.ucdavis.edu/~deloera/researchsummary/
nonreg.pdf.
[13] D. Maclagan, B. Sturmfels, Introduction to tropical geometry, to appear,
available at http://homepages.warwick.ac.uk/staff/D.Maclagan/
papers/TropicalBook.html
[14] G. Mikhalkin, Decomposition into pairs-of-pants for complex algebraic
hypersurfaces, Topology 43, 1035-1065, Elsevier, (2004), available at
http://arxiv.org/abs/math/0205011
[15] G. Mikhalkin, Real algebraic curves, the moment map and amoebas,
Annals of Mathematics 151 (2000), available at http://arxiv.org/
abs/math/0010018.
[16] G. Mikhalkin, Tropical Geometry, course notes, available at
https://www.ma.utexas.edu/rtgs/geomtop/rtg/notes/Mikhalkin_
Tropical_Lectures.pdf.
[17] M. S. Narasimhan, Several complex variables, The university of Chicago
press, Chicago and London, 1971.
[18] M. Passare, H. Rullgard, Amoebas, Monge-Ampère measures, and
triangulations of the Newton polytope, Duke Math J. volume 121
3 (2004), available at http://www2.math.su.se/reports/2000/10/
2000-10.pdf.
[19] O. Viro, Dequantization of real algebraic geometry on logarithmic paper, Proceedings of the 3rd European Congress of Mathematicians,
Birkhäuser, Progress in Math, 201, (2001), available at http://arxiv.
org/abs/math/0005163.
[20] O. Viro, Real plane algebraic curves: constructions with controlled
topology, Leningrad Mathematical Journal 1 (1990).
59
[21] J. Richter-Gebert, B. Sturmfels, T. Theobald, First Steps in Tropical
Geometry, Contemporary Mathematics 377, Amer. Math. Soc., (2005),
available at http://arxiv.org/abs/math/0306366.
[22] I. Shafarevic, Basic Algebraic Geometry 2, Springer, Berlin, 1977.
[23] W. Stein et al., Sage computer algebra, available at http://www.
sagemath.org/.
[24] B. Sturmfels, Course notes, available at http://math.berkeley.edu/
~bernd/tropical/BMS.html.
[25] T. Theobald, Computing Amoebas, Experimental Mathematics, Volume
11, issue 4 (2002), available at http://projecteuclid.org/euclid.
em/1057864661.
[26] G. M. Ziegler, lectures on Polytopes, Springer-Verlag, Berlin, 1995.
60