# A tropical characterization of algebraic subvarieties of toric varieties over non-archimedean fields

## TL;DR

This paper characterizes algebraic subvarieties of toric varieties over non-archimedean fields by their tropicalizations, showing algebraicity corresponds to tropicalizations being finite unions of polyhedra, extending known results.

## Contribution

It provides a tropical criterion for algebraicity of analytic subvarieties in toric varieties over non-archimedean fields, generalizing previous complex case results.

## Key findings

- Algebraic subvarieties have tropicalizations as finite unions of polyhedra.
- The converse, from tropicalization to algebraicity, is established.
- Extends known complex results to non-archimedean settings.

## Abstract

We study the tropicalizations of analytic subvarieties of normal toric varieties over complete non-archimedean valuation fields. We show that a Zariski closed analytic subvariety of a normal toric variety is algebraic if its tropicalization is a finite union of polyhedra. Previously, the converse direction was known by the theorem of Bieri and Groves. Over the field of complex numbers, Madani, L. Nisse, and M. Nisse proved similar results for analytic subvarieties of tori.