Ideals of polynomial semirings in tropical mathematics

TL;DR

This paper studies the structure of ideals, especially prime ideals, in polynomial semirings over layered and supertropical domains, establishing foundational theorems and characterizations relevant to tropical mathematics.

Contribution

It characterizes prime ideals in layered polynomial semirings, showing they are generated by finitely many binomials, and extends classical ideal theorems to layered tropical settings.

Findings

Prime ideals are finitely generated by binomials.

Layered tropical versions of Principal Ideal and Hilbert Basis Theorems are proved.

A detailed description of ideals arising from layered varieties is provided.

Abstract

We describe the ideals, especially the prime ideals, of semirings of polynomials over layered domains, and in particular over supertropical domains. Since there are so many of them, special attention is paid to the ideals arising from layered varieties, for which we prove that every prime ideal is a consequence of finitely many binomials. We also obtain layered tropical versions of the classical Principal Ideal Theorem and Hilbert Basis Theorem.