Bounds for the Hilbert-Kunz Multiplicity of Singular Rings

In this paper, we prove that the Watanabe-Yoshida conjecture holds up to dimension 7. Our primary new tool is a function, \(\varphi _J(R;z^t),\) 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 its rate of growth is related to the size of \(e_{\textrm{HK}}(R)\). We combine techniques from Celikbas et al. (Nagoya Math. J. 205, 149–165, 2012) and Aberbach and Enescu (Nagoya Math. J. 212, 59–85, 2013) 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.