Rational Ehrhart Theory

TL;DR

This paper introduces a unifying generating-function approach to rational Ehrhart quasipolynomials, extending classical concepts like Gorenstein polytopes to the rational setting and connecting various recent developments in lattice point enumeration.

Contribution

It presents a new generating-function ansatz for rational Ehrhart quasipolynomials and defines $b3$-rational Gorenstein polytopes, broadening the scope of Ehrhart theory.

Findings

Unified framework for classical and rational Ehrhart quasipolynomials

Introduction of $b3$-rational Gorenstein polytopes

Connections to generalized reflexive polytopes

Abstract

The Ehrhart quasipolynomial of a rational polytope $\mathsf{P}$ encodes fundamental arithmetic data of $\mathsf{P}$, namely, the number of integer lattice points in positive integral dilates of $\mathsf{P}$. Ehrhart quasipolynomials were introduced in the 1960s, satisfy several fundamental structural results and have applications in many areas of mathematics and beyond. The enumerative theory of lattice points in rational (equivalently, real) dilates of rational polytopes is much younger, starting with work by Linke (2011), Baldoni-Berline-Koeppe-Vergne (2013), and Stapledon (2017). We introduce a generating-function ansatz for rational Ehrhart quasipolynomials, which unifies several known results in classical and rational Ehrhart theory. In particular, we define $\gamma$-rational Gorenstein polytopes, which extend the classical notion to the rational setting and encompass the generalized reflexive polytopes studied by Fiset-Kasprzyk (2008) and Kasprzyk-Nill (2012).