Do most polynomials generate a prime ideal?

TL;DR

This paper investigates conditions under which polynomials with certain monomial supports generate prime ideals, providing criteria for primality and applications to tropical geometry.

Contribution

It offers necessary and sufficient combinatorial conditions for the primality of polynomial ideals based on monomial supports and Newton polytopes.

Findings

Conditions for the radical of the ideal to be prime over algebraically closed fields.

In characteristic zero, these conditions ensure the ideal itself is prime.

Application to connectedness of stable intersections of tropical hypersurfaces.

Abstract

For which monomial supports do most polynomials generate a prime ideal? We give necessary and sufficient conditions for the radical of the ideal to be prime over an algebraically closed field. In characteristic zero, the same conditions give primeness. As an application we show that under the same combinatorial conditions on Newton polytopes, the stable intersection of tropical hypersurfaces is connected through codimension one.