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Andres R. Vindas Melendez: Decompositions of Ehrhart h*-polynomials for rational
polytopes

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Abstract

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.
This is joint work with Matthias Beck (San Francisco State Univ. and FU Berlin) and
Ben Braun (Univ. of Kentucky).
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