Toric arc schemes and $q$-enumeration of lattice points

TL;DR

This paper introduces a weighted enumeration of lattice points in polytopes, connecting combinatorial weights with toric geometry, and explores the asymptotic Gaussian behavior of the generating function at q=1.

Contribution

It develops a new weighted enumeration framework linked to toric arc schemes and generalizes Rogers-Szegő polynomials with geometric insights.

Findings

Derived a Brion-type formula for the weighted generating function.

Established the asymptotic Gaussian behavior of coefficients at q=1.

Connected combinatorial weights with toric geometry structures.

Abstract

We introduce a natural weighted enumeration of lattice points in a polytope, and give a Brion-type formula for the corresponding generating function. The weighting has combinatorial significance, and its generating function may be viewed as a generalization of the Rogers-Szeg\H{o} polynomials. It also arises from the geometry of the toric arc scheme associated to the normal fan of the polytope. We show that the asymptotic behavior of the coefficients at $q = 1$ is Gaussian.