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arXiv · 2302.03761

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

Dave Anderson·Aniket Shah
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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.

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Dave Anderson, Aniket Shah. 2023-02-07. Toric arc schemes and $q$-enumeration of lattice points. https://arxiv.org/abs/2302.03761

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