A geometric interpretation of Krull dimensions of $\boldsymbol{T}$-algebras

TL;DR

This paper explores the Krull dimensions of tropical semirings and semifields, establishing a geometric interpretation related to tropical varieties and providing results on the dimension of rational function semifields of tropical curves.

Contribution

It introduces a geometric perspective on Krull dimensions of tropical algebraic structures and relates these dimensions to tropical varieties and their polyhedral complexes.

Findings

Krull dimension equals the maximum of the dimension of the tropical variety plus one.

Krull dimension of rational function semifields of tropical curves is two.

Establishes a connection between algebraic and polyhedral dimensions in tropical geometry.

Abstract

We investigate Krull dimensions of semirings and semifields dealt in tropical geometry. For a congruence $C$ on a tropical Laurent polynomial semiring $\boldsymbol{T}[X_1^{\pm}, \ldots, X_n^{\pm}]$, a finite subset $T$ of $C$ is called a finite congruence tropical basis of $C$ if the congruence variety $\boldsymbol{V}(T)$ associated with $T$ coincides with $\boldsymbol{V}(C)$. For $C$ proper, we prove that the Krull dimension of the quotient semiring $\boldsymbol{T}[X_1^{\pm}, \ldots, X_n^{\pm}] / C$ coincides with the maximum of the dimension of $\boldsymbol{V}(C)$ as a polyhedral complex plus one and that of $\boldsymbol{V}(C_{\boldsymbol{B}})$ when both $C$ and $C_{\boldsymbol{B}}$ have finite congruence tropical bases, respectively. Here $C_{\boldsymbol{B}}$ is the congruence on $\boldsymbol{T}[X_1^{\pm}, \ldots, X_n^{\pm}]$ generated by $\{ (f_{\boldsymbol{B}}, g_{\boldsymbol{B}}) \,|\, (f, g) \in C \}$ and $f_{\boldsymbol{B}}$ is defined as the tropical Laurent polynomial obtained from $f$ by replacing the coefficients of all non $-\infty$ terms of $f$ with the real number zero. With this fact, we also show that rational function semifields of tropical curves that do not consist of only one point have Krull dimension two.