F-signature functions of diagonal hypersurfaces

Abstract

Let $f$ be a diagonal hypersurface in $A_p=\mathbb{F}_p[[x_1,\dots,x_n]]$. We study the behavior of the function $\phi_{f,p}({a}/{p^e})=p^{-ne}\dim_{\mathbb{F}_p}\big(A_p/(x_1^{p^e},\dots,x_n^{p^e},f^a)\big)$ which encodes information about the F-threshold, the Hilbert-Kunz, and the F-signature functions. We prove that when $p$ goes to infinity $\phi_{f,p}$ converges to a piecewise polynomial function $\phi_f$ and the left and right derivatives of $\phi_{f,p}$ converge to $\phi'_f$. We use this fact to prove the existence of the limit F-signature and limit Hilbert-Kunz multiplicity for diagonal hypersurfaces. When $f$ is a Fermat hypersurface, we investigate the shape of the F-signature function of $f$ and provide an explicit formula for the limit F-signature and, in some cases, also for the F-signature for fixed $p$. This allows us to answer negatively to a question of Watanabe and Yoshida.