Prime congruences of additively idempotent semirings and a Nullstellensatz for tropical polynomials

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 T\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {T}}$$\end{document}, the semifield Zmax\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {Z}_{\mathrm{max}}$$\end{document} and the two element semifield B\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {B}}$$\end{document}. 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 the radical of 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 from Bertram and Easton (Adv Math 308:36–82, 2017) is proven which can be regarded as a Nullstellensatz for tropical polynomials.