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Bounds for the Hilbert-Kunz Multiplicity of Singular Rings

Ian M. Aberbach, Nicholas O Cox-Steib

Abstract

In this paper we prove that the Watanabe-Yoshida conjecture holds up to dimension $7$. Our primary new tool is a function, $\varphi_J\left(R; z^t\right),$ that interpolates between the Hilbert-Kunz multiplicities of a base ring, $R$, and various radical extensions, $R_n$. We prove that this function is concave and show that it's rate of growth is related to the size of $e_{HK}\left(R\right)$. We combine several known techniques to get effective lower bounds for $\varphi,$ which translate to improved bounds on the size of Hilbert-Kunz multiplicities of singular rings. The improved inequalities are powerful enough to show that the conjecture of Watanabe and Yoshida holds in dimension $7$.

Bounds for the Hilbert-Kunz Multiplicity of Singular Rings

Abstract

In this paper we prove that the Watanabe-Yoshida conjecture holds up to dimension . Our primary new tool is a function, that interpolates between the Hilbert-Kunz multiplicities of a base ring, , and various radical extensions, . We prove that this function is concave and show that it's rate of growth is related to the size of . We combine several known techniques to get effective lower bounds for which translate to improved bounds on the size of Hilbert-Kunz multiplicities of singular rings. The improved inequalities are powerful enough to show that the conjecture of Watanabe and Yoshida holds in dimension .
Paper Structure (16 sections, 14 theorems, 103 equations)

This paper contains 16 sections, 14 theorems, 103 equations.

Table of Contents

  1. Introduction
  2. The Watanabe-Yoshida Conjecture
  3. Main Results
  4. Setup and Notation
  5. Reducing to the Case of a Complete, Normal Domain with Algebraically Closed Residue Field
  6. Radical Extensions
  7. The Convex $\varphi_J$ Function
  8. Definition and Concavity
  9. Bounds Involving $\varphi_{J}\left(R; \, z^{t}\right)$
  10. Explicit Bounds and the Result in Dimension $7$
  11. Special Values of $e$ and $\mu$
  12. New Bounds in Dimension $7$
  13. Conclusion
  14. Higher Dimensions
  15. The $\varphi_J$ Function
  16. ...and 1 more sections

Key Result

Theorem 1.2

The Watanabe-Yoshida Conjecture holds in dimension $d$, for all $d \le 7.$

Theorems & Definitions (25)

  • Conjecture 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 1.4
  • Lemma 2.1
  • proof
  • Lemma 2.4
  • proof
  • Lemma 3.1
  • proof
  • ...and 15 more