Analysis in Hilbert-Kunz theory

TL;DR

This paper introduces new integration formulas for the $h$-function in Hilbert-Kunz theory, leading to results on asymptotic behavior, inequalities, and characterizations of invariants in local rings of characteristic p.

Contribution

It develops integration formulas for the $h$-function of hypersurfaces and applies them to extend known results and prove new inequalities in Hilbert-Kunz theory.

Findings

Asymptotic behavior of Hilbert-Kunz multiplicity for Fermat hypersurfaces of degree 3.

Validation of Watanabe and Yoshida's inequality for all odd primes.

Generalization of an inequality by Caminata, Shideler, Tucker, and Zerman.

Abstract

This paper focuses on a numerical invariant for local rings of characteristic $p$ called $h$-function, that recovers several important invariants, including the Hilbert-Kunz multiplicity, $F$-signature, $F$-threshold, and $F$-signature of pairs. In this paper, we prove some integration formulas for the $h$-function of hypersurfaces defined by polynomials of the form $\phi(f_1,\ldots,f_s)$, where $\phi$ is a polynomial and $f_i$ are polynomials in independent sets of variables. We demonstrate some applications of these integration formulas, including the following three applications. First, we establish the asymptotic behavior of the Hilbert-Kunz multiplicity for Fermat hypersurfaces of degree 3, extending the degree 2 case previously resolved by Gessel and Monsky. Second, we prove an inequality conjectured by Watanabe and Yoshida holds for all odd primes, generalizing a result of Trivedi. We give a characterization of the cases where the inequality is strict. Third, we generalize an inequality initially established by Caminata, Shideler, Tucker, and Zerman.