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

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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 j · jK the amoeba is the image of X under the map defined by LogK (z1; :::; zn) = (log(jz1jK ); ::::; log(jz1jK )). 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 discrim- inants 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 trop- ical 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 ! 1. 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 2 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 v2B λvv with B ⊆ A, λv 2 (0; 1] and v2B 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 ' is a nonzero functional then any level set is a hyperplane and for c 2 R the set fx 2 Rnj'(x) ≤ cg 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 ' 2 (Rn)_ such that F = fx 2 P j'(x) = Mg, where M is the maximum of ' on P . The face F is called the supporting face of ' and is denoted with P '. Because M is the maximum of ' on P we have F = P \ fx 2 Rnj'(x) ≥ Mg, 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 2 C then every face of P is an element of C. 2. If P; Q 2 C then P \ Q is a face of P and of Q. The support jCj of C is the union over all elements of C. We call C a polyhedral subdivision of jCj. Notice that jCj 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 ' between complexes C and D is a bijection ': jCj ! jDj 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) 2 P such that if (t; x) 2 P for some t 2 R then t ≤ λ. Thus Pmax consists of all points of P which are maximal on a line of the form R × fxg n with x 2 R .
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