Mixed Ehrhart polynomials

  • Christian Haase
  • Martina Juhnke-Kubitzke
  • Raman Sanyal
  • Thorsten Theobald
Keywords: Lattice polytope, (Mixed) Ehrhart polynomial, Discrete (mixed) volume, h^*-vector, Real roots


For lattice polytopes $P_1,\ldots, P_k \subseteq \mathbb{R}^d$, Bihan (2016) introduced the discrete mixed volume $DMV(P_1,\dots,P_k)$ in analogy to the classical mixed volume.  In this note we study the associated mixed Ehrhart polynomial $ME_{P_1, \dots,P_k}(n) = DMV(nP_1, \dots, nP_k)$.  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 $r  P_1, \ldots, rP_k$ the corresponding mixed $h^*$-polynomial has only real roots and as a consequence  the mixed $h^*$-vector becomes non-negative.


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