Let (R,mathfrak m,k) be an F-finite, unmixed, d-dimensional Noetherian local ring of characteristic p>0. If R is not regular, then its Hilbert–Kunz multiplicity satisfies e_HK(R)≥ 1+frac1d!. The related F-signature maximality conjecture asserts that, for a nonregular d-dimensional Cohen–Macaulay local ring, s(R)≤ 1-frac1d!.

Watanabe–Yoshida conjecture on F-invariants

Watanabe and Yoshida formulated the conjecture in 2005 as a lower bound for Hilbert–Kunz multiplicity, together with a related maximality conjecture for $F$ -signature. The general statement remains broader than the cases currently claimed.

Known results

In odd characteristic, the Hilbert–Kunz conjecture is known in dimensions $d\le 3$ ; dimension $4$ is also settled, with characteristic-dependent values.
For dimensions $d=5,6$ , the corresponding lower bound is known.
The associated $F$ -signature inequality remains conjectural, with partial bounds for non-Gorenstein Cohen–Macaulay domains and selected toric rings.
Low-dimensional cases for the three related statements were known in dimensions $2$ through $4$ when $p>2$ .

June 2025 claim and September 2026 extensions

A June 2025 preprint claims the strong conjecture for nonregular complete intersections in every positive characteristic, including uniqueness of the quadric minimizer; this is substantial claimed progress, not a verified resolution of the full problem. A September 2026 preprint adds formulas for limit invariants of simple singularities and Fermat hypersurfaces, without claiming completeness. A December 2025 preprint reports “another proof,” but the retrieved abstract does not identify its scope clearly.

Current status (as of October 2026): The complete-intersection case is claimed in an unverified preprint, while the full Watanabe–Yoshida conjecture remains open.

Watanabe–Yoshida conjecture on F-invariants

Primary source

Additional references

Progress summary

Solutions 0