**Contemporary Mathematics**

Volume: 605;
2013;
256 pp;
Softcover

MSC: Primary 14;

Print ISBN: 978-1-4704-1021-6

Product Code: CONM/605

List Price: $89.00

Individual Member Price: $71.20

**Electronic ISBN: 978-1-4704-1410-8
Product Code: CONM/605.E**

List Price: $89.00

Individual Member Price: $71.20

# Tropical and Non-Archimedean Geometry

Share this page *Edited by *
*Omid Amini; Matthew Baker; Xander Faber*

A co-publication of the AMS and Centre de Recherches Mathématiques

Over the past decade, it has become apparent
that tropical geometry and non-Archimedean geometry should be studied
in tandem; each subject has a great deal to say about the other.

This volume is a collection of articles dedicated to one or both of
these disciplines. Some of the articles are based, at least in part,
on the authors' lectures at the 2011 Bellairs Workshop in Number
Theory, held from May 6–13, 2011, at the Bellairs Research
Institute, Holetown, Barbados.

Lecture topics covered in this volume include polyhedral structures
on tropical varieties, the structure theory of non-Archimedean curves
(algebraic, analytic, tropical, and formal), uniformization theory for
non-Archimedean curves and abelian varieties, and applications to
Diophantine geometry. Additional articles selected for inclusion in
this volume represent other facets of current research and illuminate
connections between tropical geometry, non-Archimedean geometry, toric
geometry, algebraic graph theory, and algorithmic aspects of systems
of polynomial equations.

#### Table of Contents

# Table of Contents

## Tropical and Non-Archimedean Geometry

- Preface vii8 free
- Polyhedral structures on tropical varieties 116 free
- Lifting nonproper tropical intersections 1530
- Fewnomial systems with many roots, and an Adelic Tau Conjecture 4560
- 1. Introduction 4560
- 2. Applications and new conjectures on straight-line programs 5065
- 3. Background: from triangles to toric deformations and tropical varieties 5368
- 4. Proving our main results 5974
- 5. Wrapping up: Invariance of 𝑌_{𝐿}(𝑛,𝑘), and the proofs of Proposition 1.4, Theorem 1.10, and Lemmata 4.1 and 4.2 6378
- Acknowledgements 6782
- References 6782

- Non-Archimedean Coamoebae 7388
- On the structure of non-archimedean analytic curves 93108
- Non-archimedean uniformization and monodromy pairing 123138
- Diophantine geometry and analytic spaces 161176
- 1. Introduction 161176
- 2. The standard height function 161176
- 3. Heights for line bundles, canonical heights 164179
- 4. Level sets for the canonical height 167182
- 5. Equidistribution (arithmetic case) 169184
- 6. Measures on analytic spaces 172187
- 7. Bogomolov’s conjecture for totally degenerate abelian varieties 175190
- Acknowledgements 177192
- References 178193

- Tropicalizing vs. compactifying the Torelli morphism 181196
- Primer for the algebraic geometry of sandpiles 211226

#### Readership

Graduate students and research mathematicians interested in tropical or non-archimedean geometry and algebraic graph theory.