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, JRz^{t}, that interpolates between the Hilbert-Kunz multiplicities of a base ring, R, and various radical extensions, Rₙ. We prove that this function is concave and show that it's rate of growth is related to the size of R. We combine techniques from CHDZ and IanNewEst 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 7.