Hilbert-Kunz series, F-signature series, and weak p-fractals
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909863630602240 |
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| author | Caminata, Alessio Zerman, Francesco |
| author_facet | Caminata, Alessio Zerman, Francesco |
| contents | We extend the theory of $p$-fractals of Monsky and Teixeira by introducing the notion of weak $p$-fractal. We prove that for a hypersurface $f$ having rational Hilbert-Kunz series is equivalent to the weak $p$-fractality of the associated function $ϕ_{f,p}$ and having rational F-signature series is equivalent to the weak $p$-fractality of the reflection $\overlineϕ_{f,p}$. In addition, we prove some results characterizing the shape of the generating series of numerical functions which are quasi-polynomials in $p^n$. This is motivated by the fact that the Hilbert-Kunz and F-signature functions take this form in several examples of interest. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_19355 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hilbert-Kunz series, F-signature series, and weak p-fractals Caminata, Alessio Zerman, Francesco Commutative Algebra Number Theory We extend the theory of $p$-fractals of Monsky and Teixeira by introducing the notion of weak $p$-fractal. We prove that for a hypersurface $f$ having rational Hilbert-Kunz series is equivalent to the weak $p$-fractality of the associated function $ϕ_{f,p}$ and having rational F-signature series is equivalent to the weak $p$-fractality of the reflection $\overlineϕ_{f,p}$. In addition, we prove some results characterizing the shape of the generating series of numerical functions which are quasi-polynomials in $p^n$. This is motivated by the fact that the Hilbert-Kunz and F-signature functions take this form in several examples of interest. |
| title | Hilbert-Kunz series, F-signature series, and weak p-fractals |
| topic | Commutative Algebra Number Theory |
| url | https://arxiv.org/abs/2510.19355 |