Tropical functions on a skeleton

TL;DR

This paper establishes a finiteness property for tropical functions on skeleta within Berkovich analytifications, using a model-theoretic approach involving stable completions of algebraic varieties.

Contribution

It introduces a new finiteness result for tropical functions on skeleta, leveraging stable completions as a framework, which generalizes previous results in Berkovich spaces.

Findings

Finiteness of tropical functions on skeleta in Berkovich spaces

Extension of finiteness results to stable completions of algebraic varieties

Framework unifies model-theoretic and analytic approaches

Abstract

We prove a general finiteness statement for the ordered abelian group of tropical functions on skeleta in Berkovich analytifications of algebraic varieties. Our approach consists in working in the framework of stable completions of algebraic varieties, a model-theoretic version of Berkovich analytifications, for which we prove a similar result, of which the former one is a consequence.