# Algebraic and Combinatorial Aspects of Tropical Geometry

Share this page *Edited by *
*Erwan Brugallé; Mariá Angélica Cueto; Alicia Dickenstein; Eva-Maria Feichtner; Ilia Itenberg*

This volume contains the proceedings of the
CIEM workshop on Tropical Geometry, held December 12–16, 2011, at the
International Centre for Mathematical Meetings (CIEM), Castro
Urdiales, Spain.

Tropical geometry is a new and rapidly developing field of
mathematics which has deep connections with various areas of
mathematics and physics, such as algebraic geometry, symplectic
geometry, complex analysis, dynamical systems, combinatorics,
statistical physics, and string theory. As reflected by the content of
this volume, this meeting was mainly focused on the geometric side of
the tropical world with an emphasis on relations between tropical
geometry, algebraic geometry, and combinatorics.

This volume provides an overview of current trends concerning
algebraic and combinatorial aspects of tropical geometry through
eleven papers combining expository parts and development of modern
techniques and tools.

# Table of Contents

## Algebraic and Combinatorial Aspects of Tropical Geometry

- Dedication v6 free
- Preface ix10 free
- Intersection multiplicity numbers between tropical hypersurfaces 112 free
- Introduction 112
- 1. Toric hypersurfaces 314
- 2. Intersection of toric hypersurfaces 415
- 3. Tropical hypersurfaces 617
- 4. Intersection of tropical hypersurfaces 819
- 5. Intersection multiplicity numbers between tropical hypersurfaces 1021
- 6. Tropical intersection numbers via volumes of Cayley polytopes 1526
- 7. Non degenenerate tropical complete intersections 1627
- References 1829

- On the total curvature of tropical hypersurfaces 2132
- Tropical Teichmüller and Siegel spaces 4556
- Elliptic Curves in Honeycomb Form 8798
- Tropically unirational varieties 109120
- A Guide to Tropicalizations 125136
- 1. Introduction 125136
- 2. Analytification 127138
- 3. Tropicalization 131142
- 4. Models over the valuation ring and reduction 132143
- 5. Initial degeneration 136147
- 6. Affine toric schemes over a valuation ring 137148
- 7. Toric schemes over a valuation ring 146157
- 8. Tropical cone of a variety 150161
- 9. Projectively embedded toric varieties 152163
- 10. The Gröbner complex 158169
- 11. Compactifications in toric schemes 163174
- 12. Tropical compactifications 172183
- 13. Tropical multiplicities 176187
- 14. Proper intersection with orbits 180191
- Appendix A. Convex geometry 184195
- References 187198

- Categorical Notions ofLayered Tropical Algebra and Geometry 191202
- 1. Introduction 191202
- 2. Background 197208
- 3. Pre-ordered semigroups, monoids, and semirings 202213
- 4. Integral domains and monoids with valuation 206217
- 5. The layered structure 208219
- 6. The layered categories and the corresponding tropicalization functors 216227
- 7. The function category 221232
- 8. The tropicalization functor on polynomials and their roots 228239
- 9. The category of affine layered geometry 229240
- References 232243

- Tropical realization spaces for polyhedral complexes 235246
- Local tropicalization 253264
- 1. Introduction 254265
- 2. Geometry of semigroups 257268
- 3. Toric varieties 262273
- 4. Linear varieties associated to semigroups 264275
- 5. Valuation spaces 270281
- 6. An affine theory of tropicalization 274285
- 7. Extensions of valuations 276287
- 8. The formal toric rings 𝐾[[Γ]] 281292
- 9. Standard bases 284295
- 10. Tropical bases 291302
- 11. The local finiteness theorem 293304
- 12. Comparison between local and global tropicalization 301312
- 13. Toroidal meaning of local tropicalization 306317
- 14. An extension of the definition of tropicalization 310321
- 15. Comparison with the literature 311322
- References 313324

- Some acyclic systems of permutations are not realizable by triangulations of a product of simplices 317328
- Tropical (1,1)-homology for floor decomposed surfaces 329340