标题： 希尔伯特-库恩理论分析 显示英文标题

标题： Analysis in Hilbert-Kunz theory

摘要： 本文关注特征为$p$的局部环的一个数值不变量，称为$h$函数，该函数可以恢复多个重要的不变量，包括 Hilbert-Kunz 乘数、$F$签名、$F$阈值以及对的$F$签名。 在本文中，我们证明了由形式为$\phi(f_1,\ldots,f_s)$的多项式定义的超曲面的$h$函数的一些积分公式，其中$\phi$是一个多项式，$f_i$是独立变量集中的多项式。 我们展示了这些积分公式的应用，包括以下三个应用。 首先，我们建立了次数为 3 的费马超曲面的 Hilbert-Kunz 乘数的渐近行为，扩展了 Gessel 和 Monsky 之前解决的次数为 2 的情况。 其次，我们证明了 Watanabe 和 Yoshida 提出的不等式对所有奇素数成立，推广了 Trivedi 的结果。 我们给出了不等式严格成立的情况的特征。 第三，我们推广了 Caminata、Shideler、Tucker 和 Zerman 最初建立的不等式。 显示英文摘要

摘要： 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.