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| Natura: | Preprint |
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2025
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| Accesso online: | https://arxiv.org/abs/2507.13898 |
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| _version_ | 1866908601205915648 |
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| author | Meng, Cheng |
| author_facet | Meng, Cheng |
| contents | 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 $ϕ(f_1,\ldots,f_s)$, where $ϕ$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_13898 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Analysis in Hilbert-Kunz theory Meng, Cheng Commutative Algebra 13A35, 13H15 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 $ϕ(f_1,\ldots,f_s)$, where $ϕ$ 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. |
| title | Analysis in Hilbert-Kunz theory |
| topic | Commutative Algebra 13A35, 13H15 |
| url | https://arxiv.org/abs/2507.13898 |