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, φJ(R;zt),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi _J(R;z^t),$$\end{document} that interpolates between the Hilbert-Kunz multiplicities of a base ring, R, and various radical extensions, Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_n$$\end{document}. We prove that this function is concave and show that its rate of growth is related to the size of eHK(R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$e_{\textrm{HK}}(R)$$\end{document}. 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 φ,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi ,$$\end{document} 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.