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arXiv · 2402.05822

Bounds for the Hilbert-Kunz Multiplicity of Singular Rings

Ian M. Aberbach·Nicholas O Cox-Steib
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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 $7$.

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Ian M. Aberbach, Nicholas O Cox-Steib. 2024-02-08. Bounds for the Hilbert-Kunz Multiplicity of Singular Rings. https://arxiv.org/abs/2402.05822

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