Ehrhart Functions of Weighted Lattice Points

TL;DR

This paper explores weighted Ehrhart functions of convex polytopes, analyzing their algebraic and combinatorial properties, and establishing conditions for rationality, reciprocity, and connections to Ehrhart rings.

Contribution

It introduces three types of weighted Ehrhart functions, proves their properties, and links weighted Ehrhart rings to classical Ehrhart rings of weight lifting polytopes.

Findings

Weighted Ehrhart series can be rational functions under certain conditions.

$q$- and $s$-weighted Ehrhart reciprocity theorems are established.

Weighted Ehrhart rings correspond to Ehrhart rings of weight lifting polytopes.

Abstract

This paper studies three different ways to assign weights to the lattice points of a convex polytope and discusses the algebraic and combinatorial properties of the resulting weighted Ehrhart functions and their generating functions and associated rings. These will be called $q$-weighted, $r$-weighted, and $s$-weighted Ehrhart functions, respectively. The key questions we investigate are \emph{When are the weighted Ehrhart series rational functions and which classical Ehrhart theory properties are preserved? And, when are the abstract formal power series the Hilbert series of Ehrhart rings of some polytope?} We prove generalizations about weighted Ehrhart $h^*$-coefficients of $q$-weighted Ehrhart series, and show $q$- and $s$-weighted Ehrhart reciprocity theorems. Then, we show the $q$- and $r$-weighted Ehrhart rings are the (classical) Ehrhart rings of weight lifting polytopes.