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F-signature functions of diagonal hypersurfaces

Alessio Caminata, Samuel Shideler, Kevin Tucker, Francesco Zerman

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 $φ_{f,p}({a}/{p^e})=p^{-ne}\dim_{\mathbb{F}_p}\big(A_p/(x_1^{p^e},\dots,x_n^{p^e},f^a)\big)$ which encodes information about the F-threshold, the Hilbert-Kunz, and the F-signature functions. We prove that when $p$ goes to infinity $φ_{f,p}$ converges to a piecewise polynomial function $φ_f$ and the left and right derivatives of $φ_{f,p}$ converge to $φ'_f$. 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.

F-signature functions of diagonal hypersurfaces

Abstract

Let be a diagonal hypersurface in . We study the behavior of the function which encodes information about the F-threshold, the Hilbert-Kunz, and the F-signature functions. We prove that when goes to infinity converges to a piecewise polynomial function and the left and right derivatives of converge to . We use this fact to prove the existence of the limit F-signature and limit Hilbert-Kunz multiplicity for diagonal hypersurfaces. When is a Fermat hypersurface, we investigate the shape of the F-signature function of and provide an explicit formula for the limit F-signature and, in some cases, also for the F-signature for fixed . This allows us to answer negatively to a question of Watanabe and Yoshida.
Paper Structure (15 sections, 27 theorems, 198 equations)

This paper contains 15 sections, 27 theorems, 198 equations.

Table of Contents

  1. introduction
  2. Preliminaries
  3. Representation ring
  4. F-signature and Hilbert-Kunz of hypersurfaces
  5. Sequences in $\Gamma$ and their operators
  6. Limit $\phi$-functions for diagonal hypersurfaces
  7. The limit $\lim_{p\to\infty}\phi_{f,p}$
  8. The limit derivatives of $\phi_{f,p}$
  9. Limit $\phi$-function of the Fermat quadric
  10. F-signature series and sum of hypersurfaces
  11. F-signature function of Fermat hypersurfaces
  12. A question by Watanabe and Yoshida
  13. Explicit examples
  14. Fermat cubic in four variables
  15. Non-diagonal hypersurfaces

Key Result

Theorem A

Let $2\leq d_1\leq \cdots\leq d_n$ be integers and let $f=x_1^{d_1}+\cdots+x_n^{d_n}$. Then the functions $\phi_{f,p}$ converge uniformly, as $p$ goes to infinity, to a piecewise polynomial function $\phi_f$ given by where $C_{\lambda}(t)$ for $\lambda\in\mathbb{Z}_{\geq1}$ is the piecewise polynomial function with the sum taken over all choices of $\epsilon_0,\dots,\epsilon_n\in\{\pm1\}$ with $

Theorems & Definitions (63)

  • Theorem A: Theorem \ref{['thm_15']} and Theorem \ref{['thm16']}
  • Theorem B: Theorem \ref{['thm:shapeFermat']}
  • Lemma 2.1
  • proof
  • Remark 2.2
  • Example 2.3
  • Theorem 3.2: Han--Monsky
  • Theorem 3.3
  • proof
  • Remark 3.4
  • ...and 53 more