Introduction to Tropical Geometry

Description

Preface Chapter 1. Tropical Islands 1.1. Arithmetic 1.2. Dynamic Programming 1.3. Plane Curves 1.4. Amoebas and their Tentacles 1.5. Implicitization 1.6. Group Theory 1.7. Curve Counting 1.8. Compactifications 1.9. Exercises Chapter 2. Building Blocks 2.1. Fields 2.2. Algebraic Varieties 2.3. Polyhedral Geometry 2.4. Gröbner Bases 2.5. Gröbner Complexes 2.6. Tropical Bases 2.7. Exercises Chapter 3. Tropical Varieties 3.1. Hypersurfaces 3.2. The Fundamental Theorem 3.3. The Structure Theorem 3.4. Multiplicities and Balancing 3.5. Connectivity and Fans 3.6. Stable Intersection 3.7. Exercises Chapter 4. Tropical Rain Forest 4.1. Hyperplane Arrangements 4.2. Matroids 4.3. Grassmannians 4.4. Linear Spaces 4.5. Surfaces 4.6. Complete Intersections 4.7. Exercises Chapter 5. Tropical Garden 5.1. Eigenvalues and Eigenvectors 5.2. Tropical Convexity 5.3. The Rank of a Matrix 5.4. Arrangements of Trees 5.5. Monomials in Linear Forms 5.6. Exercises Chapter 6. Toric Connections 6.1. Toric Background 6.2. Tropicalizing Toric Varieties 6.3. Orbits 6.4. Tropical Compactifications 6.5. Geometric Tropicalization 6.6. Degenerations 6.7. Intersection Theory 6.8. Exercises Bibliography Index