Bounds on the Hilbert-Kunz multiplicity (Q2880839)

Let \((R, \mathfrak{m})\) be a \(d\)-dimensional commutative Noetherian local ring of prime characteristic \(p\). The main result of the paper is a new lower \textit{I. M. Aberbach} and \textit{F. Enescu} [Mich. Math. J. 57, 1--16 (2008; Zbl 1222.13005)], the authors of the paper introduce new ideas and also obtain other interesting results on Hilbert-Kunz multiplicity. For example, the paper presents a new inequality relating the Hilbert-Kunz multiplicity, Hilbert-Samuel multiplicity and the F-signature of the ring. Moreover, under some conditions on the ring and assuming an algebraically closed residue field, the authors prove that for an integrally closed \(\mathfrak{m}\)-primary ideal \(I\), \(\text{e}_{HK}(I) \geq \text{length}(R/I) + \text{e}_{HK}(R)-1\). \textit{I. M. Aberbach} and the reviewer have also refined their earlier work and improved the originally obtained lower bound in a new work [``New estimates of Hilbert-Kunz multiplicities for local rings of fixed dimension'', to appear in Nagoya Math. J.; \url{arXiv:1101.5078}], but the paper under review produces a better bound in important cases. The bound obtain in this paper is still weaker than the lower bound conjectured by \textit{K.-I. Watanabe} and \textit{K.-I. Yoshida} in [Nagoya Math. J. 177, 47--75 (2005; Zbl 1076.13009)].