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Analysis in Hilbert-Kunz theory

Cheng Meng

TL;DR

The paper develops a unifying framework for characteristic $p$ invariants by introducing the multivariate $h$-function and establishing integration formulas that express $h_{R,I,f}$ as RS integrals against kernel functions $D_oldsymbol{}}$, enabling computations across composite polynomials. Building on Han–Monsky representation rings and Teixeira’s $p$-fractal structure, the authors derive both fixed- and limit-characteristic formulas, prove convergence of derivatives, and provide tools to compute limit invariants like $e_{HK}$ and $s$ for families of hypersurfaces. As applications, they prove the Watanabe–Yoshida inequality for all odd primes, characterize strict inequality cases, and obtain asymptotics for Fermat hypersurfaces, alongside explicit $h$-functions for several singularities. These results yield new computational methods and conceptual links between Hilbert–Kunz theory, $F$-signature, and singularity invariants, with potential implications for understanding regularity and singularities in positive characteristic.

Abstract

This paper focuses on a numerical invariant for local rings of characteristic $p$ called $h$-function, that recovers several important invariants, including the Hilbert-Kunz multiplicity, $F$-signature, $F$-threshold, and $F$-signature of pairs. In this paper, we prove some integration formulas for the $h$-function of hypersurfaces defined by polynomials of the form $φ(f_1,\ldots,f_s)$, where $φ$ is a polynomial and $f_i$ are polynomials in independent sets of variables. We demonstrate some applications of these integration formulas, including the following three applications. First, we establish the asymptotic behavior of the Hilbert-Kunz multiplicity for Fermat hypersurfaces of degree 3, extending the degree 2 case previously resolved by Gessel and Monsky. Second, we prove an inequality conjectured by Watanabe and Yoshida holds for all odd primes, generalizing a result of Trivedi. We give a characterization of the cases where the inequality is strict. Third, we generalize an inequality initially established by Caminata, Shideler, Tucker, and Zerman.

Analysis in Hilbert-Kunz theory

TL;DR

The paper develops a unifying framework for characteristic invariants by introducing the multivariate -function and establishing integration formulas that express as RS integrals against kernel functions , enabling computations across composite polynomials. Building on Han–Monsky representation rings and Teixeira’s -fractal structure, the authors derive both fixed- and limit-characteristic formulas, prove convergence of derivatives, and provide tools to compute limit invariants like and for families of hypersurfaces. As applications, they prove the Watanabe–Yoshida inequality for all odd primes, characterize strict inequality cases, and obtain asymptotics for Fermat hypersurfaces, alongside explicit -functions for several singularities. These results yield new computational methods and conceptual links between Hilbert–Kunz theory, -signature, and singularity invariants, with potential implications for understanding regularity and singularities in positive characteristic.

Abstract

This paper focuses on a numerical invariant for local rings of characteristic called -function, that recovers several important invariants, including the Hilbert-Kunz multiplicity, -signature, -threshold, and -signature of pairs. In this paper, we prove some integration formulas for the -function of hypersurfaces defined by polynomials of the form , where is a polynomial and are polynomials in independent sets of variables. We demonstrate some applications of these integration formulas, including the following three applications. First, we establish the asymptotic behavior of the Hilbert-Kunz multiplicity for Fermat hypersurfaces of degree 3, extending the degree 2 case previously resolved by Gessel and Monsky. Second, we prove an inequality conjectured by Watanabe and Yoshida holds for all odd primes, generalizing a result of Trivedi. We give a characterization of the cases where the inequality is strict. Third, we generalize an inequality initially established by Caminata, Shideler, Tucker, and Zerman.

Paper Structure

This paper contains 37 sections, 93 theorems, 431 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Numerical invariants in characteristic $p$.
  3. Motivation and aims.
  4. Main results
  5. Verification of previous results
  6. Revealing new results
  7. The outline of the paper
  8. Preliminaries on analysis: distribution, dirac delta function, and Riemann-Stieltjes integral
  9. Riemann-Stieltjes integral
  10. The Riemann-Stieltjes integral as a distribution
  11. Integration by parts
  12. Convergence of Riemann-Stieltjes integral
  13. Multivariate $h$-function
  14. Properties of $D_{T_1+T_2}$ in characteristic $p$ and limit function
  15. Value of $D_{T_1+T_2}$ and its limit
  16. ...and 22 more sections

Key Result

Theorem 1.1

For any fixed characteristic $p\geq 3$,

Figures (12)

  • Figure 1: Position of $B_1 \sim B_4,T_0$
  • Figure 2: Explicit and Recursive cubes for IFS of $D_2$
  • Figure 3: Explicit and Recursive cubes for IFS of $D_3$
  • Figure 4: The intersection of $\Theta$ with $t_3$-planes
  • Figure 5: A demonstration of $\Theta \cap C'$. Here the gray shape represents for $C'\cap\Theta$ which is a tetrahedron, and the red segments form the $1$-skeleton of $T_0$. We see $\partial C'\cap\Theta=\partial C'\cap (C'\cap \Theta)$ is the $1$-skeleton. The endpoints of the thick blue segment falls on this $1$-skeleton.
  • ...and 7 more figures

Theorems & Definitions (221)

  • Theorem 1.1: See \ref{['7.1 WY inequality proof']}
  • Theorem 1.2: See \ref{['7.2 WY strict inequality']}
  • Proposition 1.3: See \ref{['7.2 WY strict inequality on F-sig']}
  • Corollary 1.4: See \ref{['8 corollary d=3']}
  • Definition 2.1
  • Example 2.2
  • Definition 2.4
  • Definition 2.6
  • Definition 2.7
  • Proposition 2.8: Rudinmathanalysis, Theorem 6.9
  • ...and 211 more