Supertropical Polynomials and Resultants

TL;DR

This paper advances supertropical algebra by defining resultants and relatively prime polynomials, establishing a supertropical Bézout's theorem, and extending factorization and Nullstellensatz concepts within supertropical geometry.

Contribution

It introduces supertropical resultants, a supertropical version of Bézout's theorem, and a refined factorization and Nullstellensatz framework for supertropical polynomials.

Findings

Polynomials are relatively prime iff they lack common tangible roots.

Resultants are tangible iff polynomials are relatively prime.

A supertropical version of Bézout's theorem is established.

Abstract

This paper, a continuation of [3], involves a closer study of polynomials of supertropical semirings and their version of tropical geometry in which we introduce the concept of relatively prime polynomials and resultants, with the aid of some topology. Polynomials in one indeterminant are seen to be relatively prime iff they do not have a common tangible root, iff their resultant is tangible. The Frobenius property yields a morphism of supertropical varieties; this leads to a supertropical version of B\'ezout's theorem. Also, a supertropical variant of factorization is introduced which yields a more comprehensive version of Hilbert's Nullstellensatz than the one given in [3].