Séminaire Lotharingien de Combinatoire, 85B.38 (2021), 12 pp.
Matthias Beck, Benjamin Braun and Andrés R. Vindas-Meléndez
Decompositions of Ehrhart h*-Polynomials for Rational Polytopes
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.
Received: December 1, 2020.
Accepted: March 1, 2021.
Final version: April 29, 2021.
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