Mixed Ehrhart polynomials

Abstract

For lattice polytopes P-1,...,P-k subset of R-d, Bihan (2016) introduced the discrete mixed volume DMV(P-1,...,P-k) in analogy to the classical mixed volume. In this note we study the associated mixed Ehrhart polynomial MEp(1),...,p(k)(n) = DMV(nP(1),...,nP(k)). We provide a characterization of all mixed Ehrhart coefficients in terms of the classical multivariate Ehrhart polynomial. Bihan (2016) 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 rP(1),...,rP(k) the corresponding mixed h*-polynomial has only real roots and as a consequence the mixed h*-vector becomes non-negative.