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, $φ_J\left(R; z^t\right),$ 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 $φ,$ 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 $7$.
math.AC↗