Hilbert-Kunz series, F-signature series, and weak p-fractals
Alessio Caminata, Francesco Zerman
TL;DR
The paper develops a framework linking the rationality of Hilbert-Kunz and F-signature series to weak p-fractals, introducing a relaxed fractal notion that suffices for rational generating functions. It proves that HK-series rationality for a hypersurface is equivalent to the weak p-fractality of the associated function $\phi_{f,p}$, while F-signature rationality corresponds to the weak p-fractality of its reflection $\overline{\phi}_{f,p}$. It further analyzes the generating series of quasi-polynomials in $p^n$, showing they are rational with denominators determined by cyclotomic-type factors, and studies the special quasi-polynomial form $e_n=a_dp^{dn}+a_0(n)$, including when cancellations occur via common factors with $1-z^M$. The work also provides partial classifications of when primitive cyclotomic roots appear in the numerator and highlights open questions about the full structure of these cancellations and their relation to HK/F-signature data. Overall, the results give a combinatorial and algebraic route to understanding when HK and F-signature series are rational and how quasi-polynomial shapes influence the pole structure of their generating functions.
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 $φ_{f,p}$ 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.