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 $\varphi (f_1,\dots ,f_s)$ in terms of the Riemann-Stieltjes integral, where $\varphi$ 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.

This is the revised and submitted version of the manuscript "Analysis in Hilbert-Kunz theory". The introduction has been largely rewritten. Sections 9 and 10 and Subsections 4.2 and 6.3 have been deleted. The opening parts of Subsections 7.1 and 7.2 have been relocated to Subsection 2.5. The previous Subsection 4.6 has been moved to Subsection 8.1