Refined Ehrhart series and bigraded rings

TL;DR

This paper introduces refined Ehrhart series for polytopes, computes them for specific shapes using algebraic methods, and derives new summation formulas and polynomial characterizations.

Contribution

It provides the first comprehensive algebraic computation of refined Ehrhart series for key polytopes and generalizes classical q-integer identities.

Findings

Refined Ehrhart series computed for simplices, cross-polytopes, and hypercubes.

Derived summation formulas for products of q-integers with different arguments.

Characterized a refined Eulerian polynomial algebraically.

Abstract

We study a natural set of refinements of the Ehrhart series of a closed polytope, first considered by Chapoton. We compute the refined series in full generality for a simplex of dimension d, a cross-polytope of dimension d, respectively a hypercube of dimension d<4, using commutative algebra. We deduce summation formulae for products of q-integers with different arguments, generalizing a classical identity due to MacMahon and Carlitz. We also present a characterisation of a certain refined Eulerian polynomial in algebraic terms.