Mixed Ehrhart polynomials

TL;DR

This paper explores the properties of mixed Ehrhart polynomials associated with lattice polytopes, providing new insights into their coefficients, introducing a mixed h*-vector, and proving real-rootedness and non-negativity for large dilates.

Contribution

It introduces the study of mixed Ehrhart polynomials, offers interpretations of their coefficients, and proves real-rootedness and non-negativity of the mixed h*-vector for large dilates.

Findings

Mixed Ehrhart polynomial coefficients relate to discrete mixed volumes.

For large dilates, the mixed h*-polynomial has only real roots.

The mixed h*-vector becomes non-negative for large dilates.

Abstract

For lattice polytopes $P_1,\ldots, P_k \subseteq \mathbb{R}^d$, Bihan (2014) introduced the discrete mixed volume $\mathrm{DMV}(P_1,\dots,P_k)$ in analogy to the classical mixed volume. In this note we initiate the study of the associated mixed Ehrhart polynomial $\mathrm{ME}_{P_1,\dots,P_k}(n) = \mathrm{DMV}(nP_1,\dots,nP_k)$. We study properties of this polynomial and we give interpretations for some of its coefficients in terms of (discrete) mixed volumes. Bihan (2014) showed that the discrete mixed volume is always non-negative. Our investigations yield simpler proofs for certain special cases. We also introduce and study the associated mixed $h^*$-vector. We show that for large enough dilates $r P_1, \ldots, rP_k$ the corresponding mixed $h^*$-polynomial has only real roots and as a consequence the mixed $h^*$-vector becomes non-negative.