Weighted lattice point sums in lattice polytopes, unifying Dehn--Sommerville and Ehrhart--Macdonald

Abstract

Let $V$ be a real vector space of dimension $n$ and let $M\subset V$ be a lattice. Let $P\subset V$ be an $n$-dimensional polytope with vertices in $M$, and let $φ\colon V\rightarrow \CC $ be a homogeneous polynomial function of degree $d$ (i.e., an element of $\Sym^{d} (V^{*})$). For $q\in \ZZ_{>0}$ and any face $F$ of $P$, let $D_{\phi ,F}(q)$ be the sum of $\phi$ over the lattice points in the dilate $qF$. We define a generating function $G_φ(q,y) \in \QQ [q] [y]$ packaging together the various $D_{\phi ,F}(q)$, and show that it satisfies a functional equation that simultaneously generalizes Ehrhart--Macdonald reciprocity and the Dehn--Sommerville relations. When $P$ is a simple lattice polytope (i.e., each vertex meets $n$ edges), we show how $G_{\phi }$ can be computed using an analogue of Brion--Vergne's Euler--Maclaurin summation formula.

arXiv comment19 pages, three figures; replaced old (and, so we thought, cool) title "The Bott-Brion-Dehn-Ehrhart-Euler-Khovanskii-Maclaurin-Puhklikov-Sommerville-Vergne formula for simple lattice polytopes" and corrected typos; to appear in Discrete & Computational Geometry