Intermediate Sums on Polyhedra: Computation and Real Ehrhart Theory

TL;DR

This paper introduces an algorithmic framework for intermediate sums on polyhedra, bridging integrals and discrete sums, and extends Ehrhart theory to real dilations, enabling new computational and theoretical insights.

Contribution

It develops an algorithmic theory for parametric intermediate generating functions and provides explicit formulas for real Ehrhart quasi-polynomials of polyhedra.

Findings

Algorithm for computing Ehrhart quasi-polynomials as step polynomials

Extension of Ehrhart theory to real dilations

Explicit formulas for intermediate sums on polyhedra

Abstract

We study intermediate sums, interpolating between integrals and discrete sums, which were introduced by A. Barvinok [Computing the Ehrhart quasi-polynomial of a rational simplex, Math. Comp. 75 (2006), 1449--1466]. For a given semi-rational polytope P and a rational subspace L, we integrate a given polynomial function h over all lattice slices of the polytope P parallel to the subspace L and sum up the integrals. We first develop an algorithmic theory of parametric intermediate generating functions. Then we study the Ehrhart theory of these intermediate sums, that is, the dependence of the result as a function of a dilation of the polytope. We provide an algorithm to compute the resulting Ehrhart quasi-polynomials in the form of explicit step polynomials. These formulas are naturally valid for real (not just integer) dilations and thus provide a direct approach to real Ehrhart theory.