Hilbert-Kunz multiplicity of quadrics decreases

Abstract

In this paper we prove that the Green ring of $\mathbb{Z}/p^e\mathbb{Z}$ is the $e$-fold tensor product of the Green ring of $\mathbb{Z}/p\mathbb{Z}$, and this isomorphism is given by $p$-adic expansions of integers. As an application of this isomorphism, we compute the Hilbert-Kunz function and Hilbert-Kunz multiplicity of Fermat quadrics. Then we use Gelfand transform of the Green ring to give an analytical expression of this Hilbert-Kunz multiplicity, and prove that it decreases with its characteristic, thus giving a positive answer to a conjecture of Yoshida.

Two important changes: the formula for Ehrhart polynomial of Fibonacci and extended Fibonacci polytopes has been added in subsection 4.3, and we have included computational results for the multiplicity for all dimensions up to 15 in subsection 5.1. Fixed some typos