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

TL;DR

This paper introduces two new decomposition formulas for the $h^*$-polynomial of rational polytopes, extending classical results and providing new proofs and inequalities relevant to Ehrhart theory.

Contribution

It generalizes existing decompositions for lattice polytopes to rational polytopes, offering novel proofs and inequalities for the $h^*$-polynomial.

Findings

Generalized Betke--McMullen formula for rational polytopes

Provided a new proof of Stanley’s Monotonicity Theorem for rational polytopes

Extended Stanley and Hibi inequalities to 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.