Tropical hypersurfaces and subdivisions

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Table of contents

1 Introduction 
1.1 Univariate polynomials
1.2 Sparse polynomial systems
1.2.1 Polyhedral bounds
1.2.2 Fewnomial bounds
1.3 Results prior to this thesis
1.3.1 Around Khovanskii’s bound
1.3.2 Using combinatorial patchworking
1.3.3 Systems supported on a circuit
1.3.4 Around Kuschnirenko’s conjecture
1.3.5 Around a polynomial-fewnomial conjecture
1.4 Results of the thesis
1.4.1 Chapter 3: Intersecting a sparse plane curve and a line
1.4.2 Chapter 4: Positive intersection points of a trinomial and a t-nomial curves
1.4.3 Chapter 5: Characterization of circuits supporting polynomial systems with the maximal number of positive solutions
1.4.4 Chapter 6: Constructing polynomial systems with many positive solutions
2 Preliminaries 
2.1 A brief introduction to real dessins d’enfant
2.2 A brief introduction to tropical geometry
2.2.1 Polytopes and subdivisions
2.2.2 Tropical polynomials and hypersurfaces
2.2.3 Tropical hypersurfaces and subdivisions
2.2.4 Intersection of tropical hypersurfaces
2.2.5 Generalized Viro theorem and tropical reformulation
2.2.6 Reduced systems and non-transversal intersections
3 Intersecting a sparse plane curve and a line 
3.1 Preliminary results
3.2 Proof of Theorem 3.1
3.3 Optimality
4 Positive intersection points of a trinomial and a t-nomial curves 
4.1 Introduction and statement of the main results
4.2 Proof of Theorem 4.1
4.3 Proof of Theorem 4.2
4.3.1 Reduction to a simpler case
4.3.2 Analysis of dessins d’enfant
4.3.3 End of the proof of Theorem 4.2
4.4 The case of two trinomials: proof of Theorem 4.3
4.4.1 Proof of Proposition 4.33
4.4.2 End of proof of Theorem 4.3
5 Characterization of circuits 
5.1 Technical preamble
5.2 Proof of the “only if” direction of Theorem 5.1
5.3 Proof of the “if” direction of Theorem 5.1
6 Constructing polynomial systems 
6.1 Statement of the main results
6.1.1 For normalized systems
6.1.2 Transversal intersection points
6.2 Non-transversal intersection components of type (I)
6.3 Base fans and tropical intersections
6.4 Preliminary case-by-case analysis for n = k = 2
6.4.1 Approximation polynomials for type-(I) intersections
6.4.2 Reduced systems for type-(II) intersections
6.4.3 Reduced systems for type-(III) intersections at the origin
6.4.4 Type-(III) intersections outside the origin
6.5 Proof of Theorem 6.1
6.5.1 First part of Theorem 6.48
6.5.2 Construction: second part of Theorem 6.48
6.6 Proof of Theorem 6.3 (part 1)
6.6.1 First case: 0 < α < β
6.6.2 The case α = 0 < β
6.6.3 The case α < 0 < β
6.6.4 The case α < 0 and β = 0
6.6.5 The case α < β < 0
6.6.6 The case α = β < 0
6.7 Proof of Theorem 6.3 (part 2)
6.7.1 The case 0 < α ≤ β
6.7.2 The case α < 0 < β
6.7.3 The case α ≤ β < 0
7 Introduction (en Fran¸cais) 
Bibliography