In algebra, the Hilbert–Kunz function of a local ring (R, m) of prime characteristic p is the function where q is a power of p and m[q] is the ideal generated by the q-th powers of elements of the maximal ideal m.[1] The notion was introduced by Ernst Kunz, who used it to characterize a regular ring as a Noetherian ring in which the Frobenius morphism is flat.
Watanabe and Yoshida strengthened some of Kunz's results, showing that in the unmixed case, the ring is regular precisely when c=1.
Hochster and Huneke related Hilbert-Kunz multiplicities to "tight closure" and Brenner and Monsky used Hilbert–Kunz functions to show that localization need not preserve tight closure.
A comprehensive overview is to be found in Craig Huneke's article "Hilbert-Kunz multiplicities and the F-signature" arXiv:1409.0467.
This article is also found on pages 485-525 of the Springer volume "Commutative Algebra: Expository Papers Dedicated to David Eisenbud on the Occasion of His 65th Birthday", edited by Irena Peeva.