Factorizations of tropical and sign polynomials

TL;DR

This paper investigates the factorization properties of polynomials over tropical and sign hyperfields, classifying irreducibles and analyzing unique factorization, with algorithms for division by linear factors.

Contribution

It provides a complete classification of irreducible polynomials and explores the unique factorization properties over tropical and sign hyperfields.

Findings

Tropical polynomials factor uniquely into irreducibles.

Sign polynomials do not have unique factorization.

Division algorithms for roots are established.

Abstract

In this text, we study factorizations of polynomials over the tropical hyperfield and the sign hyperfield, which we call `tropical polynomials' and `sign polynomials', respectively. We classify all irreducible polynomials in either case. We show that tropical polynomials factor uniquely into irreducible factors, but that unique factorization fails for sign polyomials. We describe division algorithms for tropical and sign polynomials by linear terms that correspond to roots of the polynomials.