Rationality of Hilbert-Kunz multiplicities: a likely counterexample
The Hilbert-Kunz multiplicity of a hypersurface is in many cases a rational number, but the author believes that \(F=uv+x^3+y^3+xyz\in {\mathbb Z}_2[x,y,z,u,v]\) has the Hilbert-Kunz multiplicity \((4/3)+(5/14\sqrt{7})\). Let \(e_{nj}\) be the colength of the ideal \((x^q,y^q,z^q,(x^3+y^3+xyz)^j)\subset {\mathbb Z}_2[x,y,z]\), where \(q=2^n\), \(n\in {\mathbb N}\). Then \(e_{n1}\) is \((7q^2-q-3)/3\) or \((7q^2-q-5)/3\) according as \(q\equiv 1\) or \(2\) mod \(3\) [see \textit{R.-O. Buchweitz} and \textit{Q. Chen}, J. Algebra 197, No. 1, 246--267 (1997; Zbl 0898.13009)]. Here it is conjectured a formula for \(e_{nj}\), which implies the above non rational value of the Hilbert-Kunz multiplicity of \(F\).
scientific article; zbMATH DE number 5604550