Factorization of polynomials in supertropical algebra

TL;DR

This paper investigates polynomial factorization in supertropical algebra, revealing where unique factorization fails due to geometric reasons and identifying specific subsets where it still holds.

Contribution

It introduces a semi-degenerate structure between classical and tropical algebra and analyzes factorization properties within this framework.

Findings

Unique factorization fails in multiple variables due to geometric reasons.

Certain subsets of polynomials maintain unique factorization.

The study bridges classical and tropical algebra through supertropical structures.

Abstract

Tropical geometry is a degeneration of classical geometry which loose the property of unique factorization for polynomials. In this paper we explore a structure that is known to be a semi-degeneration between the classical algebra and the tropical algebra, and show that unique factorization fails in several variables due to geometric reasons, not just algebraic. We also show that unique factorization does hold for a certain interesting subset of polynomials.