Finitely generated congruences on tropical rational function semifields

TL;DR

This paper characterizes when congruences on tropical rational function semifields are finitely generated, linking this property to the geometric structure of associated subsets in real space, and applies it to tropical curves.

Contribution

It provides a complete characterization of finitely generated congruences in tropical rational function semifields based on geometric conditions.

Findings

Finitely generated congruences correspond to finite unions of rational polyhedral sets.

The closure of a subset determines the finite generation of the associated congruence.

Application to the structure of tropical curves.

Abstract

We prove that the congruence on the tropical rational function semifield in $n$-variables associated with a subset $V$ of $\boldsymbol{R}^n$ is finitely generated if and only if the closure of $V$ is a finite union of $\boldsymbol{R}$-rational polyhedral sets. With this fact, we characterize rational function semifields of tropical curves.