Geodesic
Mixed Ehrhart polynomials
Christian Haase1 ; Martina Juhnke-Kubitzke2 ; Raman Sanyal3 ; Thorsten Theobald3
1FU Berlin
2Universität Osnabrück
3Goethe-Universität Frankfurt
The electronic journal of combinatorics, Tome 24 (2017) no. 1
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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.
DOI : 10.37236/5815
Classification : 52B20
Mots-clés : lattice polytope, mixed Ehrhart polynomial, discrete mixed volume, \(h^\ast\)-vector, real roots

Christian Haase  1   ; Martina Juhnke-Kubitzke  2   ; Raman Sanyal  3   ; Thorsten Theobald  3

1 FU Berlin
2 Universität Osnabrück
3 Goethe-Universität Frankfurt 
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     author = {Christian Haase and Martina Juhnke-Kubitzke and Raman Sanyal and Thorsten Theobald},
     title = {Mixed {Ehrhart} polynomials},
     journal = {The electronic journal of combinatorics},
     year = {2017},
     volume = {24},
     number = {1},
     doi = {10.37236/5815},
     zbl = {1360.52008},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/5815/}
}
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Christian Haase; Martina Juhnke-Kubitzke; Raman Sanyal; Thorsten Theobald. Mixed Ehrhart polynomials. The electronic journal of combinatorics, Tome 24 (2017) no. 1. doi: 10.37236/5815

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