Lattice points in vector-dilated quadratic irrational polytopes

TL;DR

This paper extends Ehrhart theory to quadratic irrational polytopes with vector dilations, showing that lattice point counts follow a polynomial-like behavior and establishing a reciprocity law for the leading term.

Contribution

It introduces a generalization of Ehrhart theory to polytopes with vertices in quadratic fields under vector dilations, expanding previous scalar dilation results.

Findings

Leading term of lattice point count behaves like an Ehrhart polynomial

Established Ehrhart-Macdonald reciprocity law for the leading term

Generalized previous work on scalar dilations to vector dilations

Abstract

We study the Ehrhart theory of quadratic irrational polytopes that undergo vector dilations. That is, for a given polytope with vertices in $\mathbb{Q}(\sqrt{D})$, and a different dilation factor for each facet, we show that the leading term of the lattice-point count behaves similar to an Ehrhart polynomial, generalizing previous work of Borda on scalar dilations of quadratic irrational polytopes. As a result, a form of the Ehrhart-Macdonald reciprocity law is obtained for the leading term.