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

TL;DR

This paper generalizes the concept of p-fractals to weak p-fractals, establishing equivalences between rationality of Hilbert-Kunz and F-signature series and weak p-fractality of associated functions, with implications for the shape of related generating series.

Contribution

It introduces weak p-fractals and proves their equivalence to rational Hilbert-Kunz and F-signature series, extending the theory of p-fractals.

Findings

Rational Hilbert-Kunz series corresponds to weak p-fractality of $\

ext{The shape of generating series of quasi-polynomial functions is characterized.}

Hilbert-Kunz and F-signature functions often take quasi-polynomial form in examples.

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.