Non-finite generatedness of the congruences defined by tropical varieties

TL;DR

This paper investigates the finite generation of certain congruences in tropical geometry, showing that most are not finitely generated and providing explicit generators for a specific case.

Contribution

It proves that congruences of the form E(Z) are generally not finitely generated, except in special cases, and explicitly constructs generators for the tropical standard line.

Findings

Most E(Z) congruences are not finitely generated.

Explicit minimal generators are provided for the tropical standard line.

The study clarifies the structure of tropical congruences related to tropical varieties.

Abstract

In tropical geometry, there are several important classes of ideals and congruences such as tropical ideals, bend congruences, and the congruences of the form $\mathbf E(Z)$. Although they are analogues of the concept of ideals of rings, it is not well known whether they are finitely generated. In this paper, we study whether the congruences of the form $\mathbf E(Z)$ are finitely generated. In particular, we show that when $Z$ is the support of a tropical variety, $\mathbf E(Z)$ is not finitely generated except for a few specific cases. In addition, we give an explicit minimal generating set of $\mathbf E(|L|)$ for the tropical standard line $L$.