UC Berkeley UC Berkeley Electronic Theses and Dissertations Title Tropical implicitization Permalink https://escholarship.org/uc/item/7d6845sr Author Cueto, Maria Angelica Publication Date 2010 Peer reviewed|Thesis/dissertation eScholarship.org Powered by the California Digital Library University of California Tropical Implicitization by Maria Angelica Cueto A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Mathematics in the Graduate Division of the University of California, BERKELEY Committee in charge: Professor Bernd Sturmfels, Chair Professor Martin Olsson Professor John P. Huelsenbeck Fall 2010 Tropical Implicitization Copyright 2010 by Maria Angelica Cueto 1 Abstract Tropical Implicitization by Maria Angelica Cueto Doctor of Philosophy in Mathematics University of California, BERKELEY Professor Bernd Sturmfels, Chair In recent years, tropical geometry has developed as a theory on its own. Its two main aims are to answer open questions in algebraic geometry and to give new proofs of celebrated classical results. The main subject of this thesis is concerned with the former: the solution of implicitization problems via tropical geometry. We develop new and explicit techniques that completely solve these challenges in four concrete examples. We start by studying a family of challenging examples inspired by algebraic statistics and machine learning: the restricted Boltzmann machines F(n; k). These machines are highly structured projective varieties in tensor spaces. They correspond to a statistical model encoded by the complete bipartite graph Kk;n, by marginalizing k of the n + k binary random variables. In Chapter 2, we investigate this problem in the most general setting. We conjecture a formula for the expected dimension of the model, verifying it in all relevant cases. We also study inference functions and their interplay with tropicalization of polynomial maps. In Chapter 3, we focus on the particular case F(4; 2), answering a question by Drton, Sturmfels and Sullivant regarding the degree (and Newton polytope) of the homogeneous equation in 16 variables defining this model. We show that its degree is 110 and compute its Newton polytope. Along the way, we derive theoretical results in tropical geometry that are crucial for other examples in this thesis, as well as novel computational methods. In Chapter 4, we study the first secant varieties of monomial projective curves from a tropical perspective. Our main tool is the theory of geometrical tropicalization developed by Hacking, Keel and Tevelev. Their theory hinges on computing the tropicalization of subvarieties of tori by analyzing the combinatorics of their boundary in a suitable (tropical) compactification. We enhance this theory by providing a formula for computing multiplicities on tropical varieties. We believe that this construction will give insight to understand higher secants of monomial projective curves, which are key objects in toric and birational geometry. In Chapter 5, we answer the general question of implicitization of parametric surfaces in 3-space via geometric tropicalization. The generic case, together with its higher-dimensional analog, was studied by Sturmfels, Tevelev and Yu. We address this problem for non-generic 2 surfaces. This involves understanding the combinatorics of the intersection of irreducible algebraic curves in the two-dimensional torus and explicitly resolving singularities of points in curves by blow-ups. We conclude with a brief discussion on open problems, including connections to Berkovich spaces, extension of the theory to non-archimedean valued fields and applications of tropical implicitization to classifying tropical surfaces in three-space. i To my (academic) family. ii Contents List of Algorithms iv List of Figures v List of Tables vii 1 Introduction 1 1.1 Tropical geometry . .1 1.1.1 Constant coefficient case . .2 1.1.2 Arbitrary coefficients case . .8 1.2 Hadamard products and their tropicalization . 11 1.3 Geometric tropicalization and tropical implicitization . 14 1.4 Algorithms for implicitization . 20 2 Geometry of the restricted Boltzmann machine 26 2.1 Introduction . 26 2.2 Algebraic varieties, Hadamard product and tropicalization . 28 2.3 The first secant variety of the n-cube . 32 2.4 The tropical model and its dimension . 37 2.5 Polyhedral geometry of parametric inference . 41 3 An implicitization challenge for the restricted Boltzmann machine 47 3.1 Introduction . 47 3.2 Geometry of the model . 49 3.3 Tropicalizing the model . 50 3.4 Newton polytope of the defining equation . 59 3.4.1 Vertices and facets . 59 3.4.2 Computing vertices . 61 3.4.3 Implementation . 65 3.4.4 Certifying facets . 66 3.4.5 Completing the polytope . 68 iii 4 Tropical secant graphs of monomial curves 72 4.1 Introduction . 72 4.2 The master graph . 74 4.3 Combinatorics of monomial curves . 79 4.4 The master graph under Hadamard products . 86 4.5 The Newton polytope of the secant hypersurface in P4 ............ 95 4.6 Chow polytopes, tropical secant lines, toric arrangements and beyond . 101 5 Implicitization of surfaces via geometric tropicalization 107 5.1 Introduction . 107 5.2 Tropical elimination and tropical implicitization . 109 5.3 Tropical implicitization for generic surfaces . 116 5.4 Tropical implicitization for non-generic surfaces . 127 5.5 Further remarks . 134 Bibliography 139 iv List of Algorithms 3.1 Ray-shooting algorithm. 62 3.2 Walking algorithm. 63 3.3 Facet certificate algorithm. 67 3.4 Approximation of a polytope by a subpolytope. 70 5.1 Tropical implicitization for generic surfaces. 138 v List of Figures 1.1 A tropical surface in R3 described as a collection of two-dimensional cones in R3 or as a non-planar graph in S2......................... 13 1.2 Intersection complex on a first compactification of X ⊂ P2 and on a resolution of this compactification with CNC. 19 1.3 Ray-shooting algorithm in dimension two. 21 2.1 Graphical representation of the restricted Boltzmann machine. 27 2.2 Quartet trees associated to the flattenings for n = 4. 35 2.3 Partitions of the 3-cube that define non-empty cones on which Φ is linear. 37 1 2.4 Subdivisions of the 3-cube that represent vertices and facets of T M3 ..... 43 1 2.5 The polyhedral complex associated to the tropical model T M3 ........ 44 1 2.6 Parameterization of T M2 ............................. 46 3.1 The restricted Boltzmann machine F(4; 2). 47 3.2 Ray-shooting and walking algorithms combined. 64 3.3 Walking from vertex to vertex in the Newton polytope. 65 3.4 Approximation algorithm. 69 4.1 The building blocks of the master graph..................... 75 4.2 The master graph associated to the curve (1 : t30 : t45 : t55 : t78). 77 4.3 Compactification of Z in Pn, with exponents f0; 30; 45; 55; 78g......... 80 4.4 Two affine charts describing the singularities of the embedding of the surface (1 − λ, !30 − λ, !45 − λ, !55 − λ, !78 − λ)..................... 81 4.5 Resolution diagram of a binomial arrangement at the origin. 82 4.6 Resolution diagram of a binomial arrangement at infinity. 82 4.7 The tropical secant graph and the Gr¨obnertropical secant graph of the mono- mial curve (1 : t30 : t45 : t55 : t78) in P4...................... 96 4.8 The Gr¨obner tropical secant graphs of two rational normal curves. 103 4 4.9 The first tropical secant complex of the line Rh(0; i1; i2; i3; i4)i in TP ..... 105 5.1 Newton polytopes of the polynomials f1; f2 and f3............... 122 5.2 Normal fans of the Newton polytopes P1; P2 and P3.............. 122 vi 5.3 Minkowski sum of P1, P2 and P3, and a strictly simplicial refinement of its normal fan. 123 5.4 Pairwise Minkowski sums of the Newton polytopes P1; P2; P3......... 124 5.5 Tropical graph of a generic surface in T3..................... 124 5.6 Newton polytopes of the polynomials f1; f2 and f3............... 125 5.7 Normal fans of the Newton polytopes P1; P2 and P3.............. 126 5.8 Minkowski sum of P1, P2 and P3, and a strictly simplicial refinement of its normal fan. 127 5.9 Tropical graph of a generic surface associated to three generic nodal curves. 128 5.10 Three Newton polytopes in R2 and common refinement of their normal fans. 128 5.11 Tropical graph of the surface parameterized by (5.11). 129 5.12 Newton polytope and dual graph of a degree 3 surface in C3.......... 129 5.13 Boundary of the closure of X in P2 and its resolution. 132 5.14 Tropical graph of a non-generic surface associated to three nodal curves. 133 5.15 Affine charts with u = 1 and t = 1 and the boundary divisors on each chart. 133 5.16 Resolution diagrams at (1 : 1 : 1) and (0 : 1 : 0). 134 5.17 Newton polytope and dual graph of a non-generic surface in C3........ 135 vii List of Tables 2.1 Special cases where Conjecture 2.2.2 holds. 45 2 3.1 Facet orbit sizes of the Newton polytope of V4 ................. 60 viii Acknowledgments Hej och hopp h¨arska jag g¨oralite roliga mattegrejer! Alex Engstr¨om I would like to express my gratitude to my advisor, Bernd Sturmfels, for all he has taught me, for the many interesting questions he posed, for helping me turn them into this thesis, and for his insights and encouragement along the way. He has generously afforded me a great deal of personal attention and thoughtful mentorship, taking a long-term view of my career. Second, I would like to thank Enrique Tobis for his support and care for the past eight years, for listening and for keeping me sane throughout the decision-making process. I would like to thank my co-authors Shaowei Lin, Jason Morton, Bernd Sturmfels, Enrique Tobis and Josephine Yu for interesting discussions, productive collaborations, and permission to include some co-authored material as part of my dissertation.