Non-finite generatedness of the congruences defined by tropical varieties

TL;DR

The paper investigates finite generation of congruences of the form $\mathbf{E}(Z)$ on tropical function semirings, focusing on whether $\mathbf{E}(|X|)$ is finitely generated when $X$ is a tropical variety. It proves a sharp criterion: $\mathbf{E}(|X|)$ is finitely generated if and only if $|X|$ is a rational affine linear subspace; otherwise, in general, $\mathbf{E}(|X|)$ is not finitely generated, with the notable exception of cases where $|X|$ is a linear subspace. The authors develop a method to show non-finite generation via Newton polytopes and unbounded directions, and they reduce ambient dimension when necessary to apply the criterion. As a concrete illustration, they provide an explicit minimal generating set for $\mathbf{E}(L)$ where $L$ is the standard tropical line in $\mathbb{R}^2$. Overall, the work clarifies when tropical congruences admit finite presentations and supplies explicit generators in key cases, advancing the tropical analogue of Noetherian behavior in algebraic geometry.

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 $L$.

Figures (1)

• Figure 1: $\Delta(\text{white circled points}) = \Delta(\text{black circled points})$

Theorems & Definitions (77)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Definition 2.3
• Lemma 2.4
• Lemma 2.5