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.