Counting lattice points in free sums of polytopes

TL;DR

This paper provides a method to compute the Ehrhart polynomial of the free sum of two lattice polytopes using combinatorial properties, generalizing previous work and linking lattice point enumeration in dilates of polytopes.

Contribution

It introduces a new approach to compute Ehrhart polynomials of free sums of polytopes via the multiplicativity of the weighted h*-polynomial, extending prior results.

Findings

Ehrhart polynomial of free sums can be computed from individual polytopes.

Weighted h*-polynomial is multiplicative under free sum.

Counting lattice points in dilates relates to free sums of polytopes.

Abstract

We show how to compute the Ehrhart polynomial of the free sum of two lattice polytopes containing the origin $P$ and $Q$ in terms of the enumerative combinatorics of $P$ and $Q$. This generalizes work of Beck, Jayawant, McAllister, and Braun, and follows from the observation that the weighted $h^*$-polynomial is multiplicative with respect to the free sum. We deduce that given a lattice polytope $P$ containing the origin, the problem of computing the number of lattice points in all rational dilates of $P$ is equivalent to the problem of computing the number of lattice points in all integer dilates of all free sums of $P$ with itself.