A guide to tropicalizations

TL;DR

This paper develops a generalized theory of tropicalizations over arbitrary non-archimedean valued fields, connecting algebraic and convex geometry through toric schemes and non-archimedean analysis.

Contribution

It extends fundamental tropical geometry results to more general fields by constructing toric schemes over valuation rings of rank 1.

Findings

Generalization of tropicalization results to arbitrary non-archimedean fields

Development of a theory of toric schemes over valuation rings

Application of non-archimedean analysis techniques

Abstract

Tropicalizations form a bridge between algebraic and convex geometry. We generalize basic results from tropical geometry which are well-known for special ground fields to arbitrary non-archimedean valued fields. To achieve this, we develop a theory of toric schemes over valuation rings of rank 1. As a basic tool, we use techniques from non-archimedean analysis.