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

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

to appear in "The Landscape of Commutative Algebra", editors M. Rossi, N.V. Trung, B. Ulrich, and J. Verma, Lecture Note Series of the London Mathematical Society, Cambridge University Press