Hilbert-Kunz multiplicity of quadrics via Ehrhart theory

TL;DR

This paper establishes that the Hilbert-Kunz multiplicity of d-dimensional quadrics in characteristic p is a rational function derived from Ehrhart polynomials, revealing its decreasing nature with dimension and its dependence on characteristic.

Contribution

It introduces a novel connection between Hilbert-Kunz multiplicity and Ehrhart theory, providing explicit rational formulas and analyzing their behavior across dimensions and characteristics.

Findings

Hilbert-Kunz multiplicity is a rational function of p for quadrics.

The multiplicity decreases as the dimension increases.

It recovers known results about the behavior of multiplicity with respect to characteristic.

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.