Prime congruences of idempotent semirings and a Nullstellensatz for tropical polynomials

TL;DR

This paper introduces a new concept of prime congruences in additively idempotent semirings, characterizes them in tropical polynomial semirings, and establishes a Nullstellensatz-like result for tropical polynomials.

Contribution

It defines prime congruences using twisted products, describes their structure in tropical polynomial semirings, and proves a Nullstellensatz for tropical polynomials, advancing tropical algebraic geometry.

Findings

Prime congruences exhibit properties analogous to prime ideals in rings.

Minimal primes correspond to monomial orderings and relate to Newton polytopes.

A Nullstellensatz for tropical polynomials is established.

Abstract

A new definition of prime congruences in additively idempotent semirings is given using twisted products. This class turns out to exhibit some analogous properties to the prime ideals of commutative rings. In order to establish a good notion of radical congruences it is shown that the intersection of all primes of a semiring can be characterized by certain twisted power formulas. A complete description of prime congruences is given in the polynomial and Laurent polynomial semirings over the tropical semifield ${\pmb T}$, the semifield $\mathbb{Z}_{max}$ and the two element semifield $\mathbb{B}$. The minimal primes of these semirings correspond to monomial orderings, and their intersection is the congruence that identifies polynomials that have the same Newton polytope. It is then shown that every finitely generated congruence in each of these cases is an intersection of prime congruences with quotients of Krull dimension $1$. An improvement of a result of A. Bertram and R. Easton from 2013 is proven which can be regarded as a Nullstellensatz for tropical polynomials.