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arXiv · 2508.17915

Hilbert-Kunz multiplicity of quadrics via Ehrhart theory

Igor Pak·Boris Shapiro·Ilya Smirnov·Ken-ichi Yoshida
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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.

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Igor Pak, Boris Shapiro, Ilya Smirnov, Ken-ichi Yoshida. 2025-08-25. Hilbert-Kunz multiplicity of quadrics via Ehrhart theory. https://arxiv.org/abs/2508.17915

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