New estimates of Hilbert-Kunz multiplicities for local rings of fixed dimension (Q2873891)

The authors present some new lower bounds for the Hilbert-Kunz Multiplicity of an unmixed nonregular ring. They start by proving a portion of a conjecture of \textit{K.-i. Watanabe} and \textit{K.-i. Yoshida} [Nagoya Math J. 177, 47--75 (2005; Zbl 1076.13009)] which states that for a formally unmixed nonregular local ring \((R, \mathfrak{m}, \overline{\mathbb{F}}_p)\) of dimension \(d\), the Hilbert-Kunz multiplicity of \(R\) is bounded below by the Hilbert-Kunz multiplicity of the hypersurface \(\overline{\mathbb{F}}_p[[x_0,\ldots,x_d]]/(x_0^2+\cdots+x_d^2)\) which in turn is bounded below by \(1+m_d\) where \(m_d\) is the coefficient of \(x^d\) in the power series representing \(\sec x+ \tan x\). Watanabe and Yoshida have proved this conjecture for rings of dimension less than or equal to 4. \textit{F. Enescu} and \textit{K. Shimomoto} [J. Algebra 285, No. 1, 222--237 (2004; Zbl 1080.13011)] have proved the first inequality for complete intersections and Yoshida has proved the second inequality for dimensions 5 and 6. In the current work, the authors use volume estimates to prove the conjecture when the Hilbert Samuel multiplicity of \(R\) is 3, 4 or 5 and when the dimension of \(R\) is 5 or 6. The remainder of the paper deals with exhibiting lower bounds for the Hilbert-Kunz multiplicity of Gorenstein rings which are not complete intersections.