Linear and Rational Factorization of Tropical Polynomials

TL;DR

This paper introduces an efficient algorithm for factorizing a broad class of tropical polynomials in multiple variables, addressing the NP-Complete challenge for bivariate cases and connecting to applications in economics and combinatorics.

Contribution

It provides a novel, intrinsic characterization of regular mixed subdivisions of integral polytopes, enabling polynomial-time factorization for certain tropical polynomials.

Findings

Efficient factorization algorithm for a rich class of tropical polynomials

Characterization of regular mixed subdivisions of integral polytopes

Connections to open problems in discrete geometry

Abstract

Already for bivariate tropical polynomials, factorization is an NP-Complete problem. In this paper, we give an efficient algorithm for factorization and rational factorization of a rich class of tropical polynomials in $n$ variables. Special families of these polynomials have appeared in economics, discrete convex analysis, and combinatorics. Our theorems rely on an intrinsic characterization of regular mixed subdivisions of integral polytopes, and lead to many open problems of interest in discrete geometry.