Lattice-point generating functions for free sums of convex sets

TL;DR

This paper derives formulas for generating functions of lattice points in free sums of convex sets, extending known results and providing conditions for product formulas in Ehrhart theory.

Contribution

It introduces new formulas for lattice point generating functions of free sums of convex sets, generalizing Braun's Ehrhart series product formula and establishing necessary and sufficient conditions.

Findings

Formulas for generating functions of lattice points in free sums.

Conditions for Braun's Ehrhart series product formula to hold.

Extension of known Ehrhart series results to broader convex sets.

Abstract

Let $\J$ and $\K$ be convex sets in $\R^{n}$ whose affine spans intersect at a single rational point in $\J \cap \K$, and let $\J \oplus \K = \conv(\J \cup \K)$. We give formulas for the generating function {equation*} \sigma_{\cone(\J \oplus \K)}(z_1,..., z_n, z_{n+1}) = \sum_{(m_1,..., m_n) \in t(\J \oplus \K) \cap \Z^{n}} z_1^{m_1}... z_n^{m_n} z_{n+1}^{t} {equation*} of lattice points in all integer dilates of $\J \oplus \K$ in terms of $\sigma_{\cone \J}$ and $\sigma_{\cone \K}$, under various conditions on $\J$ and $\K$. This work is motivated by (and recovers) a product formula of B.\ Braun for the Ehrhart series of $\P \oplus \Q$ in the case where $\P$ and $\Q$ are lattice polytopes containing the origin, one of which is reflexive. In particular, we find necessary and sufficient conditions for Braun's formula and its multivariate analogue.