Hilbert-Kunz multiplicity of quadrics via Ehrhart theory
Commutative Algebra
2026-03-25 v5 Algebraic Geometry
Combinatorics
Abstract
We show that the Hilbert-Kunz multiplicity of the d-dimensional non-degenerate quadric hypersurface of characteristic p > 2 is a rational function of p composed from the Ehrhart polynomials of integer polytopes. In consequence, we prove that the Hilbert-Kunz multiplicity of quadrics of fixed characteristic is a decreasing function of dimension and recover results of Trivedi and Gessel-Monsky on the behaviour of said Hilbert-Kunz multiplicity as a function of characteristic.
Keywords
Cite
@article{arxiv.2508.17915,
title = {Hilbert-Kunz multiplicity of quadrics via Ehrhart theory},
author = {Igor Pak and Boris Shapiro and Ilya Smirnov and Ken-ichi Yoshida},
journal= {arXiv preprint arXiv:2508.17915},
year = {2026}
}
Comments
Improved Corollary 3.12 and corrected the proof of 3.18