Analysis in Hilbert-Kunz theory

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.