Multivariate volume, Ehrhart, and $h^*$-polynomials of polytropes

TL;DR

This paper develops methods from toric geometry to compute multivariate volume, Ehrhart, and $h^*$-polynomials of lattice polytropes, providing new combinatorial descriptions and characterizations in low dimensions.

Contribution

It introduces algorithms for multivariate polynomial computation of polytropes and offers a complete combinatorial description for 3D cases, with partial results in 4D.

Findings

Algorithms for multivariate polynomials of polytropes

Complete combinatorial description in 3D

Partial characterization in 4D

Abstract

The univariate Ehrhart and $h^*$-polynomials of lattice polytopes have been widely studied. We describe methods from toric geometry for computing multivariate versions of volume, Ehrhart and $h^*$-polynomials of lattice polytropes, which are both tropically and classically convex. These algorithms are applied to all polytropes of dimensions 2,3 and 4, yielding a large class of integer polynomials. We give a complete combinatorial description of the coefficients of volume polynomials of 3-dimensional polytropes in terms of regular central subdivisions of the fundamental polytope. Finally, we provide a partial characterization of the analogous coefficients in dimension 4.