The Tropical Division Problem and the Minkowski Factorization of Generalized Permutahedra

TL;DR

This paper characterizes the factorization of tropical polynomials and extends these results to Minkowski factorizations of polytopes, providing new tools for polytope decomposition and analysis of polymatroids and Coxeter matroid polytopes.

Contribution

It introduces a characterization and construction method for tropical polynomial factorization and applies it to Minkowski polytope factorization, including a basis for polytopal fans.

Findings

Established a criterion for tropical polynomial factorization.

Developed a polytope factorization basis for polytopal fans.

Analyzed the factorization of polymatroids and Coxeter matroid polytopes.

Abstract

Given two tropical polynomials $f, g$ on $\mathbb{R}^n$, we provide a characterization for the existence of a factorization $f= h \odot g$ and the construction of $h$. As a ramification of this result we obtain a parallel result for the Minkowski factorization of polytopes. Using our construction we show that for any given polytopal fan there is a polytope factorization basis, i.e. a finite set of polytopes with respect to which any polytope whose normal fan is refined by the original fan can be uniquely written as a signed Minkowski sum. We explicitly study the factorization of polymatroids and their generalizations, Coxeter matroid polytopes, and give a hyperplane description of the cone of deformations for this class of polytopes.