Hilbert-Kunz series, F-signature series, and weak p-fractals

Alessio Caminata; Francesco Zerman

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arXiv · 2510.19355

Hilbert-Kunz series, F-signature series, and weak p-fractals

Alessio Caminata·Francesco Zerman

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Abstract

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 $\phi_{f,p}$ and having rational F-signature series is equivalent to the weak $p$-fractality of the reflection $\overline{\phi}_{f,p}$. 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.

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Alessio Caminata, Francesco Zerman. 2025-10-22. Hilbert-Kunz series, F-signature series, and weak p-fractals. https://arxiv.org/abs/2510.19355

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