Introduction to Amoebas and Tropical Geometry Milo Bogaard 17-2-2015

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 6 1.1 Polyhedral geometry ...... 6 1.2 The Newton polytope and its subdivisions ...... 8 1.3 Tropical algebra ...... 11 1.4 Tropical hypersurfaces ...... 13 1.5 The Newton polytope of a tropical curve ...... 15

2 The amoeba of a planar curve 19 2.1 Deﬁnitions and examples ...... 19 2.2 Laurent series ...... 21 2.3 The amoeba and the Newton polytope ...... 23 2.4 The components of the complement of the amoeba ...... 26

3 Puiseux series and Kapranov’s theorem 27 3.1 Valuations and Puiseux series ...... 27 3.2 Tropical polynomials ...... 29 3.3 Kapranov’s theorem ...... 32

4 A limit of amoebas 36 4.1 The construction of the limit ...... 36 4.2 The Hausdorﬀ distance ...... 38 4.3 A neighborhood of the tropical curve ...... 38 4.4 The limit ...... 40

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 Rull- gard 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 +u2w4 +u4w3 +ru3w +r2u2w2 +ruw3 +ru3w3 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 + u2w4 + u4w3 + 3.8u3w + 12.9u2w2 + 4.55uw3 + 4.6u3w3. 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.

Proposition 1.2. If A ⊂ Rn is a ﬁnite set then conv(A) consists of all points P P of the form v∈B λvv 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. 1This 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.

n Example 1.6. Let P ⊂ R × R 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} n with x ∈ R . 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[z1, ..., zn, z1 , ..., zn ]. Thus a Laurent polynomial is a finite sum f = P a zv, with a ∈ K and zv = zv1 ··· zvn . This gives the following v∈Zk v v 1 n polytope. Definition 1.8. If f = P a zv is a Laurent polynomial then the Newton v∈Zn v n polytope Newt(f) is the convex hull of the set {v ∈ Z |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(fg) = Newt(f) + Newt(g).

8 P i P i P i Proof. If f = i aiz and g = i biz , then fg = i ciz , where ci = P n j,k:j+k=i ajbk. 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(fg) ⊂ 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(fg). 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 = aibj 6= 0, so k ∈ Newt(fg).

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 = λi(v(i), i) with λi ∈ [0, 1] and P P i∈A i∈A λi = 1, so π(x) = i∈A λii ∈ Q. Thus π(P ) ⊂ Q. It is clear that π is injective. Now let x ∈ Q. We have Q = conv(A) because A contains P all vertices of Q. Thus we can write x = λii with λi ∈ [0, 1] and P P i∈A i∈A λi = 1. Now y = i∈A λi(v(i), i) ∈ P . Thus Pmax must contain an element y˜ such that y1 = y˜1,...,yn = y˜n. Clearly π(y) = π(y˜) = 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, for example f(t) = N 2 − t2. Now we deﬁne v : Q ∩ Zn → Q by Pn v(i) = j=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 ∈ Q ∩ Zn}) in R × Rn for every k ∈ Q ∩ Zn. P n Suppose that k = i∈A λii with A ⊂ Q ∩ Z and λi ∈ (0, 1]. We get

n n X X X X v(k) = f( λiij) ≥ λif(ij) j=1 i∈A j=1 i∈A n X X X = λi f(ij) = λiv(i) i∈A j=1 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 + u2w3 + u3w2 + tu + tu2 + tw + tw2 + tu3w +tuw3 + t2uw + t2u2w + t2uw2 + t2u2w2, then Newt(f) = conv({(0, 0), (0, 3), (3, 0), (3, 2), (2, 3)}). We define v : Newt(f)∩ 2 i1 i2 Z → R by v(i) = valt(ai) where ai is the coefficient of f at u w . 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 mono- mials 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.

x y Example 1.14. For t ∈ R>1 let Rt = R ∪ {−∞}, let x ⊕t y = logt(t + t ) −∞ 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 x y example. We have max{x, y} = limt→∞ logt(t + t ). To see this let x = y−x x y max{x, y}, so log(1 + t ) ≤ log(2) is bounded. Now limt→∞(t + t ) = log(1+ty−x) x+limt→∞ log(t) = x. This gives an immediate proof that T is a semiring. The process of taking the limit of the semirings Rt is called Maslov dequantization. It is explained brieﬂy on page 28 of [14].

L i1 in Deﬁnition 1.16. A tropical polynomial is a formal sum i∈I ai x1 ... xn n where I ⊂ Z is a ﬁnite index set, ai ∈ T and x1, ..., xn are real variables.

n i i1 in If x = (x1, ..., xn) and i = (i1, ..., in) ∈ Z we use x instead of x1 ... xn and i · x = i1x1 + ... + inxn, where · is the standard inner product. L i Using the definition of ⊕ and a tropical polynomial f = i∈I ai x n defines a function on R 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 n is given by maxi∈I {ai +i·x} and k ∈ I define Ck = {x ∈ R |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 ◦ c 3. The intersection k∈I (Ck ) .

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 ∈ I, so the second set is contained in the third. T ◦ c Suppose p ∈ k∈I (Ck ) , 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). n For A ⊆ I let CA = {x ∈ R |ai + i · x = f(x) for all i ∈ A}. Notice that C{k} = Ck and CA = ∩k∈ACk. 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 n in V(f) as in V(fred) as f and fred define the same function on R . Thus V(fred) ⊆ V(f). n For all A ⊆ I have CA = ∩k∈ACk and Ck = {x ∈ R |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∈ACi ∩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=jHi ∩ 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

n ∨ NF (P ) = {ϕ ∈ (R ) |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

n NF (P ) = {x ∈ R |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

n NF (P ) = {x ∈ R |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 polyghe- dron 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.

n 4. If F1, ..., Fn are faces of P then there is a face K such that ∩i=1NFi (P ) = NK (P ). We can use normal cones to ﬁnd the upper faces of a polytope. Let n n P ⊂ R × R be a polytope and let H1 = {1} × R .

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. (1,x) If NF (P ) ∩ H1 6= ∅ pick (1, x) ∈ NF (P ) ∩ H1, then F ⊆ P 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 n boundary of P is contained in ∪i=1Li. n Let F be a nonempty upper face of P then F = ∪i=1F ∩ 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 k k interior of ∩i=1Li, so there is an ε > 0 such that z + (ε, 0, ..., 0) ∈ ∩i=1Li. 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 n via Hj = {(s, y) ∈ R × R |ts + x · y ≤ M}. As w · (t, x) = M and 1 (z + (ε, 0, ..., 0)) · (t, x) > M it follows that t > 0. Now (1, t x) also deﬁnes 1 1 Hj, so (1, t x) is maximal on F . Thus (1, t 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 n defines a polytope P = conv{(ai, i)|i ∈ I} ⊂ R×R 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 Sturm- fels 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} × n V(f)| = {1} × R . 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 n NF (P ) = {(t, x) ∈ R × R |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. ◦ ◦ (1,x) Let k ∈ I such that Ck 6= ∅, pick x ∈ Ck and let F = P 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∈AN(ak,k)(P ) ∩ H1 = (∩k∈AN(ak,k)(P )) ∩ H1. By lemma 1.23 there is a face

F of P such that NF (P ) = ∩k∈AN(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 −1 −1 K[z1, ..., zn, z1 , ..., zn ], 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. ∗ n n If K has an absolute value then we have a map LogK :(K ) → R 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. −c If a = 0 then w = b so the amoeba is the set {(x, log(|c|) − log(|b|)) : x ∈ 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 2 part of R>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 de- ﬁned over the complex numbers by a Laurent polynomial f. To understand

21 the complement of the amoeba we need to look at the Laurent series expansion of 1 . A Laurent series S centered at 0 is a formal sum P a zv, where f v∈Zn v v v1 vn n av ∈ C and z = z1 ··· zn . Because Z 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 + 1 be the polynomial form example 2.3 and consider the Laurent series S = P (−1)i+ji+juiwj, where we i,j∈Z2 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 1 geometric series so it follows that S is a Laurent expansion of f .. Applying the binomial theorem to the absolute value gives

∞ X i + j X i + j X |(−1)i+j uiwj| = |u|i|w|j = (|u| + |w|)n, j j i,j∈Z2 i,j∈Z2 n=0 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 −i + 1 X −j + 1 (−1)−i+1 uiwj and (−1)−j+1 uiwj j i i,j∈Z2 i,j∈Z2

1 are Laurent expansions of f 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 1 consistent way to ﬁnd certain Laurent expansions of f 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). Proposition 2.5. (a) If S(z) = P a zv is a Laurent series centered at v∈Zn v −1 0 the domain of convergence is of the form Log (B) where B ⊂ Rn is a convex open subset. 1

1In 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.

n Corollary. If f is a Laurent polynomial then the components of R −Log(Cf ) are convex and the components correspond bijectively to the Laurent series 1 expansions of f centered at 0.

n 1 Proof. Let A be a component of R − Log(Cf ) then f is a holomorphic function deﬁned on Log−1(A). By part (b) of proposition 2.5 there is a 1 −1 unique Laurent series S which converges to f on Log (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. 0 1 If S is another Laurent series of f centered at 0, then it has a domain of −1 convergence of the form Log (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 S0 = S by part (b) of proposition 2.5.

1 Remark. It is possible compute some of the Laurent series of f directly. If γ γ is a vertex of Newt(f) then we can write f(z) = aγz (1 + g(z)), where P av v−γ g(z) = k z . Using the geometric series we get a formal identity v∈Z ,v6=γ aγ ∞ 1 X = a−1z−γ (−1)ig(z)i. (2.2.1) f(z) γ i=0 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 1 need to use Laurent series. The proofs in [8] use the Laurent series of f and work in all dimensions. For a vertex γ of Q = Newt(f) the normal cone Nγ(Q) can be identiﬁed n n with a subset of R by formula 1.5.1, so Nγ(Q) = {a ∈ R |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). n ∗ n For any v ∈ (Q − {γ}) ∩ Z and z ∈ (C ) 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 v X γ M M γ | avz | < |av||z |e ≤ k max{|av| : v 6= 0}e |z |. v6=γ v6=γ

M This shows that if we pick M such that k max{av : v 6= γ}e ≤ |aγ| we get P v γ | v6=γ avz | < |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 M k max{|av| : v 6= 0}e ≤ |aγ| where k+1 is the number of nonzero coefficients of f. 1 Remark. If we compute the cones in the example 2.3 we can use M = log( 2 ) 1 1 1 so we can pick b(0,0) = (log( 2 ), log( 2 )), b(1,0) = and b(0,1) = (0, log( 2 )).

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 2 δ lie on the y-axis, Q lies in R≥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.

2 ∗ 2 2 C which restricts to Cf in (C ) . On the w-axis of C 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)∩ 2 H1 = ∅ = Nδ(Q) ∩ H2 then Nγ(Q) + Nδ(Q) = R , 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 1 of f 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 c every component of Af . The same function was constructed by Mikhalkin in [15] using the topological linking number. The results can be stated as follows.

c Theorem 2.8 (Fosberg, Passare, Tsikh). The order of a component of Af 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 ex- 1 n pansions of f is at most the number points in Newt(f) ∩ Z and at least the number of vertices.

The following property of the order will be useful later on.

Proposition 2.9. Let k ∈ Zn∩Newt(f) and let z ∈ (C∗)n such that Log(z) ∈/ k P j c Af and |akz | > | j6=k ajz |, then the order of the complement of Af that contains Log(z) is k.

2See 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 − val(a) is a valuation on K the function defined by |a|K = e defines a non- Archimedean 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 on the field C((t)) of formal Laurent series over C defined P i by for p = i∈I ait 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 extended by the same definition. P i A Puiseux series over C is a series of the form i∈I ait , 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 i 1/k common denominator for I then i∈I ait ∈ C((t )). This shows we have 1 ∞ 1 k S k C((t )) ⊂ 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 ait 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 − val(p) absolute value on K is given by |p|K = e , so the logK function on ∗ n (K ) 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.

∗ 2 Example 3.5. Let Cf be the curve in (K ) deﬁned by f = u + w + t and 2 let T be the tropical curve V (x ⊕ y ⊕ −1). We claim LogK (Cf ) = T ∩ Q . ∗ 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 2 in T . As val(K) = Q we get LogK (Cf ) ⊂ T ∩ Q . We have

2 T ∩ Q = {(−q, −1)|q ∈ Q≥1} ∪ {(−1, −q)|q ∈ Q≥1} ∪ {(q, q)|q ∈ Q≥−1}.

−q −q −q −q For q ∈ Q<−1 we have (t , −t −t) ∈ Cf and LogK (t , −t −t) = (q, −1). −q −q −q −q For q ∈ Q>−1 we have (t , −t − t) ∈ Cf and LogK (t , −t − t) = (q, q). −q −q −q −q For q ∈ Q≤−1 we have (−t −t, t ) ∈ Cf and LogK (t , −t −t) = (−1, q). 2 Thus LogK (Cf ) = T ∩ Q .

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. i1 in j1 jn If f = ⊕i∈I ai x1 · · · xn and g = ⊕j∈J bj x1 · · · xn we put k1 kn f ⊕ g = ⊕k∈K (ak ⊕ bk) x1 · · · xn where K = I ∪ J and ak = −∞ if k1 kn k∈ / I. We also put f g = ⊕k∈K ck x1 · · · xn where K = I + J and ck = ⊕i,j;i+j=kai 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. 0 If v ∈ V (f) then there are distinct k, k ∈ I such that ak + k · v and 0 ak0 + k · v are maximal. If bl + l · v is a maximal term of g at v then 0 ak + bl + (k + l)v and ak0 + bl + (k + 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+ k bj} be the coeﬃcient of f g at x . 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 0 and j ∈ J such that ai0 + i · x < ai + i · x for every i ∈ I distinct from i and 0 0 aj0 + j · x < aj + j · x for every j ∈ J distinct from j. As in the proof of lemma 3.8 it follows that i + j = k and ck = ai + bj. 0 0 If ck = ai0 + bj0 for i ∈ I − {i} and j ∈ J − {j} then

0 0 ck + k · v = ai0 + bj0 + (i + j ) · v < ai + bj + (i + j) · v = ck + k · v, so ck is indeed unique.

P i n Deﬁnition 3.10. If K is a ﬁeld with a valuation, f = i∈I aiz with I ⊂ Z a Laurent polynomial then

M i trop(f) = − val(ai) x . i∈I

It is necessary to use − val because if ai = 0 then − val(ai) = −∞, so the i term − val(ai) x 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(fg) = x2 ⊕ −2 x ⊕ −2 and trop(f) trop(g) = x2 ⊕ −1 x ⊕ −2. However both trop(fg) 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(fg) deﬁnes the same function on Rn as trop(f) trop(g). In particular we have V (trop(fg)) = V (trop(f)) ∪ V (trop(g)).

P i P i n Proof. Let f = i∈I aiz and g = i∈J bjz with I,J ⊂ Z , then fg = P k P k∈K c˜kz with K = I + J andc ˜k = i,j;i+j=k aibj. Thus trop(fg) = L k L 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(aibj)}. i,j;i+j=k i,j;i+j=k

31 By lemma 3.2 we get

− val(c ˜k) ≤ − min {val(aibj)} = max {− val(aibj)} = ck. i,j;i+j=k i,j;i+j=k

Thus trop(fg)(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(fg)(v) ≥ − val(c ˜k) + k · v = trop(f) trop(g)(v). This shows trop(fg) 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 i Theorem 3.15 (Kapranov). If f = i∈I aiz is a polynomial over the ﬁeld n of Puiseux series in n variables then Af = V (trop(f)) ∩ Q . P i n Proof. Let f = i∈I aiz with I ⊂ Z . i For p ∈ Cf let v = LogK (p) = (− val(p1), ..., − val(pn)) then − val(aip ) = − val(ai) + i · v for all i ∈ I. We have

trop(f)(v) = max{− val(ai) + i · v}. (3.3.1) i∈I and this maximum is attained for only one index in I if and only if the k minimum mini∈I {val(aip )} is attained by one index in I. If that is the case P i k we get val( i∈I aip ) = mini∈I {val(aip )} by lemma 3.2. This is impossible P i k as ∞ = val(0) = val( i∈I aip ) and val(aip ) < ∞ for some i ∈ I. Thus we n have Af ⊆ V (trop(f)) ∩ Q . 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) = (t−v1 sα1 , ..., t−vn sαn ) defines an inclusion ϕ: K∗ → (K∗)n. If we define ψ : R → Rn by ψ(λ) = v + λα then the diagram

− val K∗ / R ϕ ψ (3.3.2)

LogK (K∗)n / 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) is the polynomial g = ϕ∗f given by g(z) = f(t−v1 zα1 , ..., t−vn zαn ) on K∗. P k To get the tropicalization of g we expand it so g(z) = k∈K bkz , where

33 P −i·v bk = i∈I:α·i=k ait . 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 com- mute with pullbacks. Example 3.16. Let f = tuw2+(−t+t2)u2w+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) = t2z3+tz5, 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}. ∗ 1 Thus trop(ϕ f) is locally linear at 0 and the root is at x = − 2 . Thus with 1 1 lemma 3.14 we get a root p of f with LogK (p) = (− 2 , − 2 ). 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 ajz be a Laurent polynomial, where Q ⊂ Z is the set n {j ∈ Z |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 n n and let Af be the amoeba of f. For r ∈ R>1 we define Logr : C → R by Logr(z1, ..., zn) = (logr |z1|, ..., logr |zn|), then Logr(Cf ) = Af ·M, where M is 1 the diagonal matrix with entries log(r) . Thus Logr(Cf ) is homeomorphic with P v(j) j Af . The coefficients of f are changed by the formula fr = j∈Q ajr z 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) = j∈Q ajt z ∈ 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 non- archimedean 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 diagonal matrix with entries log(r) . It is easy to see these set converge to the 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).

7 Figure 4.1: The set Ar with r = e 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 ) ≤ 2 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.

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 n set B ⊂ R such that limr→∞ dH (Ar,B) = 0 if such a set exists. This set is unique by the following lemma.

n Lemma 4.3. If {Ar}r∈R>1 converges to B then B consists of all b ∈ R 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 ε Ar we have an M ∈ R>0 such that dH (Ar,B) ≤ 2 for all r > M. Thus also sup d(a, B) ≤ ε . By the triangle inequality we get that d(b, A ) > ε for a∈Ar 2 r 2 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 r n contain Ar. Let M ∈ R>1 and let AK be the subset of R described by the inequalities

ck + k · x ≤ max {cj + j · x} + logr(M), j∈Q−{k} r where k ranges over Q and cj = v(j) + logr |aj|. To ensure both AK ⊂ AK r 2 and Ar ⊂ AK we pick M = max{N,Mf }, where N = #Q − 1 and Mf = −1 maxj∈Q{|aj|, |aj| }.

r Lemma 4.4. The set AK is a neighborhood of AK for every r ∈ R>1 and r limr→∞ dH (AK , 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) + r logr |aj| + j · y} + logr(Mf ). This shows AK ⊂ AK for every r ∈ R>0 and this also implies sup (d(y,Ar )) = 0. y∈AK K r Now we estimate d(y, AK ) for y ∈ AK . 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}. r By deﬁnition of AK 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 0 3 logr(M) 0 0 0 y = y− 2 (k−j1). We have v(k)+k·y ≤ v(j1)+j1 ·y , so y does not lie ||k−j1|| c 0 in the same component of AK as y. Thus d(y, AK ) ≤ d(y, y ) = 3 logr(M). This shows sup r d(y, AK ) ≤ 3 log (M), which proves the lemma. y∈AK r r We only need the sets AK to show the convergence of the Ar and the following two properties are used in the proof of theorem 4.7.

r Lemma 4.5. For every r ∈ R>0 we have Ar ⊂ AK . ∗ n Proof. If x ∈ Ar we can pick z ∈ (C ) with fr(z) = 0 and Logr(z) = x. P v(j) j v(k) k P v(j) j We have j∈Q ajr z = 0, so |akr z | = | j∈Q−{k} ajr z | for every k ∈ Q. Applying logr and the triangle inequality gives

X v(j) j ck + k · Logr(z) = logr(| ajr z |) j∈Q−{k} X v(j) j ≤ logr( |ajr z |) j∈Q−{k}

X cj j = logr( |r ||z |) j∈Q−{k}

cj j ≤ logr(N max |r ||z |) j∈Q−{k}

= max {cj + j · Logr(z)} + logr(N), j∈Q−{k}

r so x = Logr(z) ∈ AK . r Lemma 4.6. If the components of the complement of AK 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 + cj +j·x v(j) j j · x} + logr(M) an let Logr(z) = x. We have r = r |aj||z |, so we get rv(ki)|a ||zki | > M max {rv(j)|a ||zj|} > | P rv(j)a zj| by ki j∈Q−{ki} j j∈Q−{ki} j 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 Haus- dorﬀ 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 r r and limr→∞ dH (AK , AK ) = 0 we can find a T > 0 such that AK does not contain any ball of diameter ε. r Now let x ∈ AK . Because the components of the complement of AK are convex we can find a ball B of diameter ε which intersects two distinct r components C1 and C2 of the complement of AK . 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 3 3 and note that Ar is the amoeba of u + w + ruw + 1 multiplied coordinate 1 c wise with log(r) . Now theorem 4.7 shows that Af has a bounded component 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. 1This 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.

n P i Proof. Let I = Newt(g) ∩ Z and let f((z)) = i∈I z . 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 . c Thus for suﬃciently large r it follows that Ar has at least #I components. P v(i) i c The amoeba of g((z)) = i∈I r z is homeomorphic to Ar so it follows that c Ag 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), (3, 2)} and maximal number of compo- 2 P j1 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

Figure 4.3: Converging amoebas.

42 with suﬃciently large r, for example f14.

3 3 2 3 3 2 2 2 3 f14 = u + w + 1 + u w + u w + 14u + 14u + 14w + 14w + 14u w +14uw3 + 196uw + 196u2w + 196uw2 + 196u2w2

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 −1 −1 ∗ ideals of C[u, w, u , w ] is {(u − λ1, w − λ2)|λ1, λ2 ∈ C }. The correspondence between maximal ideals and points is given by mapping a maximal ∗ 2 ideal m to the single element of {(λ1, λ2) ∈ (C ) |f(λ1, λ2) = 0 for all f ∈ m}, thus (u − λ1, w − λ2) 7→ (λ1, λ2). ∗ 2 For M ∈ Mat2(Z) and α = (α1, α2) ∈ (C ) we deﬁne

M m11 m21 m12 m22 (α1, α2) = (α1 α2 , α1 α2 ). 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 ),

∗ 2 with M ∈ Mat2(Z) and α = (α1, α2) ∈ (C ) , 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 homomor- phismsϕ ˜: C[u, w] → A such thatϕ ˜(u) andϕ ˜(w) are units. The set of units i j ∗ is {αiju w |αij ∈ C and i, j ∈ Z}. Thus a homomorphismϕ ˜: C[u, w] → A m11 m21 extends to an endomorphism of A if and only ifϕ ˜(u) = α1u w and m12 m22 ∗ ϕ˜(w) = α2u w with α1, α2 ∈ C and m11, m21, m12, m22 ∈ Z. In this caseϕ ˜ 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) ∈ ∗ 2 m11 m21 m12 m22 (C ) then ϕ(u) = α1u w and ϕ(w) = α2u w , 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 βM and αMN = (αN )M ∗ 2 for all α, β ∈ (C ) and M,N ∈ Mat2(Z). This way we get a homomorphism ∗ 2 ∗ 2 GL(2, Z) → AutGrp((C ) ) and thus a semidirect product (C ) 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 oGL(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 ∗ 2 β, α ∈ (C ) and N,M ∈ Mat2(Z) then we have

m11 m21 m11n11+m21n12 m11n21+m21n22 ψ ◦ ϕ(u) = α1β1 β2 u w (5.1.2) and

m12 m22 m12n11+m22n12 m12n21+m22n22 ψ ◦ ϕ(w) = α2β1 β2 u w . (5.1.3) This shows ψ ◦ ϕ is the endomorphism deﬁned by (αβM ,NM). In particular we have Ψ(β, N)◦Ψ(α, M) = Ψ(αβM ,NM) = 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 NM 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 ∗ 2 the semigroups (C ) 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 ∗ 2 ∗ α ∈ (C ) 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−1w−1] and let msp(A) be the maximal ideal spectrum of A. If ϕ in an endomorphism of A the canonical mapϕ ˜: msp(A) → msp(A) is given byϕ ˜(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 msp(A) / (C∗)2

ϕ˜ ϕ∗

Z msp(A) / (C∗)2 ∗ commutes. Let m ∈ msp(A), then m = (u − λ1, w − λ2) for some λ1, λ2 ∈ C , ∗ M −1 M so Z(m) = (λ1, λ2) = λ and ϕ Z(m) = αλ . If f ∈ ϕ (m) then f(αλ ) = 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 ∗ 2 by α ∈ (C ) 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 AMB = 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 AMB = 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 ∗ 2 −1 −1 previous proposition. If Cf ⊂ (C ) is the curve defined by f ∈ C[u, w, u , w ] ∗ then the restriction ϕ |Cϕ(f) : Cϕ(f) → Cf is surjective. ∗ ∗ 2 Proof. We have f(ϕ (λ)) = ϕ(f)(λ) for every λ ∈ (C ) , 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 ∗ 2 α ∈ (C ) and M ∈ Mat2(Z), then Newt(ϕ(f)) = M Newt(f) Proof. The monomial uiwj occurs in f if and only if the monomial ϕ(uiwj) = un11i+n12jwn21i+n22j 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.

∗ 2 −1 −1 Corollary. Let Cf ⊂ (C ) be the curve deﬁned by f ∈ C[u, w, u , w ] 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 = MAϕ(f) − Log(α).

∗ Proof. By proposition 5.5 the map ϕ |Cϕ(f) : Cϕ(f) → Cf is surjective and by proposition 5.3 we have ϕ∗(λ) = αλM . Let T : R2 → R2 be deﬁned by x x log |α | T = M · − 1 , then it is easy to check that the diagram y y log |α2|

ϕ∗ Cϕ(f) / Cf

Log Log T R2 / 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 draw some amoebas in two 3 5 diﬀerent ways. Consider f = u + w + 1 and M = , then ϕ(f) = 7 2 u3w7 + u5w2 + 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 u3w7 + u5w2 + 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 u3w7 + u5w2 + 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 var(’xmin’,’xmax’,’ymin’,’ymax’) 2 xmin=−3 3 xmax=3 4 ymin=−3 5 ymax=3 6 var(’angle step ’,’grid s t e p ’ ) 7 g r i d s t e p =250 8 a n g l e s t e p =120 9 var(’u’,’w’) 10 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 L i s t x =[] 3 for j in range(1,grid s t e p ) : 4 r=xmin+j ∗ x g r i d s t e p s i z e 5 for k in range(0,angle s t e p ) : 6 theta=(k/angle s t e p )∗ pi ∗ I 7 z1=eˆ r ∗e ˆ( theta ) 8 g=f.substitute(u=z1) 9 S=g.roots(w, ring=ComplexField(prec=10) 10 , multiplicities=False) 11 f o r p2 in S : 12 logp2=log ( abs ( p2 ) ) 13 i f logp2>ymin and logp2

We can now plot the list of points with the following command. 1 l i s t plot(Lx, size=2,aspect 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 − u2w − w2u − 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.

1Om technische redenen kijken we alleen naar oplossingen die niet op de co¨ordinaatassen liggen. 2We 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 reëlegetallen 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 twee- dimensionale 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 + y3 + x2y4 + x4y3 + 3.8x3y + 12.9x2y2 + 4.55xy3 + 4.6x3y3 = 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

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 Kapra- nov.

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