The Watanabe–Yoshida minimal Hilbert–Kunz multiplicity conjecture

Let $d\geq 1$ and let $p>2$ be prime. Define

$$R_{p,d}=\left(\overline{\mathbb{F}_p}[x_0,\ldots,x_d]/\left\langle\sum_{i=0}^{d}x_i^2\right\rangle\right)_{\langle x_0,\ldots,x_d\rangle}.$$

Let $R$ be a $d$-dimensional unmixed local ring of characteristic $p$ with residue field $\overline{\mathbb{F}_p}$, and let $e_{HK}(R)$ denote its Hilbert–Kunz multiplicity.

Watanabe–Yoshida conjecture. If $R$ is not regular, then

$$e_{HK}(R)\geq e_{HK}(R_{p,d}),$$

with equality if and only if $R$ is formally isomorphic to $R_{p,d}$, meaning

$$\widehat{R}\simeq\widehat{R_{p,d}}.$$

The conjecture seeks the non-regular rings with the smallest possible Hilbert–Kunz multiplicity. The source identifies it as an open problem and records the cited formulation as due to Watanabe and Yoshida.