The strong form of the conjecture for complete intersections can be obtained after refinements of the results in Enescu and Shimomoto’s solution of the complete intersection case in characteristic \(p>2\). As a corollary, we also obtain the result for (formally unmixed) singularities of dimension up to 7:

Theorem (2025). Let \((R,\mathfrak{m},k)\) be an unmixed nonregular local ring of characteristic \(p>0\). Then, if the dimension of \(R\) is less than 7, or if \(R\) is a complete intersection, then \(e_\text{HK}(R)= e_\text{HK}(R_{p,d})\) implies that that \(\widehat{R} \cong k[[x_0,\dots,x_d]]/(x_0^2+\dots+x_d^2)\).

The most technical part of the proof is showing that the Hilbert–Kunz multiplicity of an \(A_2\) singularity is strictly greater than the Hilbert–Kunz multiplicity of an \(A_1\) singularity in every characteristic and dimension. This is shown via Han and Monsky’s machinery to study the Hilbert–Kunz functions of the diagonal hypersurfaces.

In characteristic 2, the main subtlety, as it is often the case, is the fact that the theory of quadratic forms differs from the case of characteristic \(\neq 2\). Namely, the standard form of a nondegenerate quadratic form, or in other words, the equation of an \(A_1\) singularity, changes slightly: if the dimension \(d\) is odd, then the singularity is defined by the equation \(Q_d := x_0x_1+...+x_{d-1}x_{d}\), and if \(d\) is even, then by \(Q_d := x_0x_1+...+x_{d-2}x_{d-1}+x_d^2\).

Theorem (2022). Let \((R,\mathfrak{m},k)\) be an unmixed nonregular local ring of characteristic 2. Then, if the dimension of \(R\) is less than 7, or if \(R\) is a complete intersection, then \(e_\text{HK}(R)\geq e_\text{HK}(R_{2,d})\), with equality only if \(\widehat{R} \cong k[[x_0,\dots,x_d]]/(Q_d)\).

See the classification of the ADE singularities by Greuel and Kröning: Simple Singularities in positive characteristic.

The result below is in this preprint on limit F-invariants. Also, see this post for context on reductions mod \(p\).

Via reduction mod \(p\), one can state the Watanabe–Yoshida conjecture in the characteristic zero case for the limit Hilbert–Kunz multiplicity. Indeed, this result can also be settled for all complete intersections and all singularities of dimension up to 7 by reduction mod \(p\), as well as the strong form of the conjecture due to the fact that also the limit Hilbert–Kunz multiplicity of an \(A_1\) and an \(A_2\) singularity also differs:

Theorem (Watanabe–Yoshida conjecture in characteristic zero). Let \(K\) be a field of characteristic \(0\), and \(R_{0,d} := K[x_0,\dots,x_d]/(x_0^2+\cdots+x_d^2).\) Let \((R,\mathfrak m,K)\) be a \(d\)-dimensional \(K\)-algebra of essentially finite type. Moreover, assume that \(R\) is unmixed and nonregular. If either \(2\le d\le 7\), or \(R\) is a complete intersection of dimension \(\ge 2\), then either \(\liminf_{p\to\infty} e_{\mathrm{HK},p}(R) > e_{\mathrm{HK},\infty}(R_{0,d}),\) or \(R\) is an \(A_1\) singularity.

Notation. This definition of lim-inf Hilbert–Kunz multiplicity is based on recent work by Liu and Pande on positivity of the limit \(F\)-signature for klt pairs: we define \(e_{\mathrm{HK},p}(R)\) as the infimum of the Hilbert–Kunz multiplicities of the characteristic \(p\) models of \(R\) (the closed fibers of a family of reductions mod \(p\)); for \(p\gg0\) it does not depend on the chosen family. We say \(R\) is an \(A_1\) singularity if \(\widehat R\cong K[[x_0,\dots,x_d]]/(a_0x_0^2+\cdots+a_dx_d^2)\) with \(a_i\in K\) nonzero.