Varieties of prime tropical ideals and the dimension of the coordinate semiring

TL;DR

This paper investigates the structure of prime tropical ideals and their associated varieties, establishing that prime ideals have at most one point in their variety and relating the dimension of tropical varieties to their coordinate semirings.

Contribution

It provides new insights into the relationship between ideals, congruences, and dimensions in tropical algebraic geometry, specifically for prime ideals and their varieties.

Findings

Prime ideals in tropical polynomial semirings have at most one point in their variety.

The dimension of an affine tropical variety is related to the dimension of its coordinate semiring.

The paper clarifies the structure of prime tropical ideals and their geometric implications.

Abstract

In this note we study the relationship between ideals and congruences of the tropical polynomial and Laurent polynomial semirings. We show that the variety of a non-zero prime ideal of the tropical (Laurent) polynomial semiring consists of at most one point. We also prove a result relating the dimension of an affine tropical variety and the dimension of its coordinate semiring.