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can be transcendental). 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. The question of how c behaves as the characteristic goes to infinity (say for a hypersurface defined by a polynomial with integer coefficients) has also received attention; once again open questions abound.
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238:. If d is the dimension of the local ring, Monsky showed that f(q)/(q^d) is c+O(1/q) for some real constant c. This constant, the "Hilbert-Kunz" multiplicity", is greater than or equal to 1. Watanabe and Yoshida strengthened some of Kunz's results, showing that in the unmixed case, the ring is regular precisely when c=1.
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and
Teixeira have treated diagonal hypersurfaces and various related hypersurfaces. But there is no known technique for determining the Hilbert–Kunz function or c in general. In particular the question of whether c is always rational wasn't settled until recently (by Brenner—it needn't be, and indeed
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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
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Hilbert–Kunz functions and multiplicities have been studied for their own sake. Brenner and
Trivedi have treated local rings coming from the homogeneous co-ordinate rings of smooth projective curves, using techniques from algebraic geometry. Han,
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E. Kunz, "On noetherian rings of characteristic p," Am. J. Math, 98, (1976), 999–1013. 1
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