Frobenius-Poincaré function and Hilbert-Kunz multiplicity

Abstract

We generalize the notion of Hilbert-Kunz multiplicity of a graded triple (M, R, I) in characteristic p > 0 by proving that for any complex number y, the limit)j e −iyj/p n exists. We prove that the limiting function in the complex variable y is entire and name this function the Frobenius-Poincaré function. We establish various properties of Frobenius-Poincaré functions including its relation with the tight closure of the defining ideal I; and relate the study Frobenius-Poincaré functions to the behaviour of graded Betti numbers of R I [p n ] as n varies. Our description of Frobenius-Poincaré functions in dimension one and two and other examples raises questions on the structure of Frobenius-Poincaré functions in general.