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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| Acceso en línea: | https://arxiv.org/abs/2403.12863 |
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| _version_ | 1866916166619889664 |
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| author | Caminata, Alessio Shideler, Samuel Tucker, Kevin Zerman, Francesco |
| author_facet | Caminata, Alessio Shideler, Samuel Tucker, Kevin Zerman, Francesco |
| contents | Let $f$ be a diagonal hypersurface in $A_p=\mathbb{F}_p[[x_1,\dots,x_n]]$. We study the behavior of the function $ϕ_{f,p}({a}/{p^e})=p^{-ne}\dim_{\mathbb{F}_p}\big(A_p/(x_1^{p^e},\dots,x_n^{p^e},f^a)\big)$ which encodes information about the F-threshold, the Hilbert-Kunz, and the F-signature functions. We prove that when $p$ goes to infinity $ϕ_{f,p}$ converges to a piecewise polynomial function $ϕ_f$ and the left and right derivatives of $ϕ_{f,p}$ converge to $ϕ'_f$. We use this fact to prove the existence of the limit F-signature and limit Hilbert-Kunz multiplicity for diagonal hypersurfaces. When $f$ is a Fermat hypersurface, we investigate the shape of the F-signature function of $f$ and provide an explicit formula for the limit F-signature and, in some cases, also for the F-signature for fixed $p$. This allows us to answer negatively to a question of Watanabe and Yoshida. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_12863 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | F-signature functions of diagonal hypersurfaces Caminata, Alessio Shideler, Samuel Tucker, Kevin Zerman, Francesco Commutative Algebra 13A35, 13D40, 14G17, 14B05 Let $f$ be a diagonal hypersurface in $A_p=\mathbb{F}_p[[x_1,\dots,x_n]]$. We study the behavior of the function $ϕ_{f,p}({a}/{p^e})=p^{-ne}\dim_{\mathbb{F}_p}\big(A_p/(x_1^{p^e},\dots,x_n^{p^e},f^a)\big)$ which encodes information about the F-threshold, the Hilbert-Kunz, and the F-signature functions. We prove that when $p$ goes to infinity $ϕ_{f,p}$ converges to a piecewise polynomial function $ϕ_f$ and the left and right derivatives of $ϕ_{f,p}$ converge to $ϕ'_f$. We use this fact to prove the existence of the limit F-signature and limit Hilbert-Kunz multiplicity for diagonal hypersurfaces. When $f$ is a Fermat hypersurface, we investigate the shape of the F-signature function of $f$ and provide an explicit formula for the limit F-signature and, in some cases, also for the F-signature for fixed $p$. This allows us to answer negatively to a question of Watanabe and Yoshida. |
| title | F-signature functions of diagonal hypersurfaces |
| topic | Commutative Algebra 13A35, 13D40, 14G17, 14B05 |
| url | https://arxiv.org/abs/2403.12863 |