Hilbert-Kunz multiplicity of quadrics via Ehrhart theory
Igor Pak, Boris Shapiro, Ilya Smirnov, Ken-ichi Yoshida
TL;DR
This work determines the Hilbert--Kunz multiplicity of the d-dimensional non-degenerate quadric hypersurface $A_{p,d}$ in characteristic $p>2$ as a rational function of $p$ built from Ehrhart polynomials of the Fibonacci polytope $F_d$ and its extended version $E_{d-2}$. It achieves this via the Han--Monsky algorithm, reformulated in a linear-algebra framework using matrices $T_a$ and $N_a$, yielding the closed form $e_{HK}(A_{p,d}) = 1 + rac{2^d F_digl( rac{p-3}{2}igr)}{p^d - E_{d-2}igl( rac{p-1}{2}igr)}$, and showing that $e_{HK}(A_{p,d})$ is a rational function in $p^2$ with degree $igl floor d/2 igr floor$ in the numerator and denominator. The results imply that for fixed $d$, $e_{HK}(A_{p,d})$ decreases with the dimension and recovers known asymptotics $e_{HK}(A_{p,d}) o 1 + { m A}_d/d!$ as $p o \/infty$, while providing an algorithmic method to compute the rational function. The approach strengthens and complements previous work by Trivedi and Gessel–Monsky on the characteristic-free and asymptotic behavior of HK multiplicities for quadrics.
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.