Limits of $F$-invariants and Riemann-Stieltjes integral

Abstract

This paper proves several results on $F$-invariants of Fermat hypersurfaces, including the proof of an inequality on the Hilbert-Kunz multiplicity of Fermat quadric hypersurfaces conjectured by Watanabe and Yoshida, the asymptotic behavior of the Hilbert-Kunz multiplicity for Fermat cubic hypersurfaces, and a strict inequality of the $F$-signature of a Fermat hypersurface whose degree is equal to its dimension. To address the above problems, this paper introduces a numerical invariant for local rings of characteristic $p$ called multivariate $h$-function. It is a real function of several variables that recovers both the Hilbert-Kunz multiplicity and the $F$-signature of hypersurface rings. We prove the above results by developing integral formulas for the $h$-function of hypersurfaces defined by polynomials of the form $\phi(f_1,\ldots,f_s)$ in terms of the Riemann-Stieltjes integral, where $\phi$ is a polynomial and $f_i$'s are polynomials in independent sets of variables, and explore how taking derivatives and taking limit of the characteristic interact with the integrals.