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arXiv · 2006.10076

Decompositions of Ehrhart $h^*$-polynomials for rational polytopes

Matthias Beck·Benjamin Braun·Andrés R. Vindas-Meléndez
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Abstract

The Ehrhart quasipolynomial of a rational polytope $P$ encodes the number of integer lattice points in dilates of $P$, and the $h^*$-polynomial of $P$ is the numerator of the accompanying generating function. We provide two decomposition formulas for the $h^*$-polynomial of a rational polytope. The first decomposition generalizes a theorem of Betke and McMullen for lattice polytopes. We use our rational Betke--McMullen formula to provide a novel proof of Stanley's Monotonicity Theorem for the $h^*$-polynomial of a rational polytope. The second decomposition generalizes a result of Stapledon, which we use to provide rational extensions of the Stanley and Hibi inequalities satisfied by the coefficients of the $h^*$-polynomial for lattice polytopes. Lastly, we apply our results to rational polytopes containing the origin whose duals are lattice polytopes.

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Matthias Beck, Benjamin Braun, Andrés R. Vindas-Meléndez. 2020-06-17. Decompositions of Ehrhart $h^*$-polynomials for rational polytopes. https://arxiv.org/abs/2006.10076

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