Frobenius-Poincaré function and Hilbert-Kunz multiplicity

We generalize the notion of Hilbert-Kunz multiplicity of a graded triple ( M , R , I ) (M,R,I) in characteristic p > 0 p>0 by proving that for any complex number y y , the limit lim n → ∞ ( 1 p n ) dim ⁡ ( M ) ∑ j = − ∞ ∞ λ ( ( M I [ p n ] M ) j ) e − i y j / p n {equation*} {n → ∞ }{lim }(1/p^n)dim(M)∑ j= -∞∞λ ( ( {M}{I[p^n]M})_j )e-iyj/p^n {equation*} exists. We prove that the limiting function in the complex variable y 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 I ; and relate the study Frobenius-Poincaré functions to the behaviour of graded Betti numbers of R I [ p n ] {R}{I[p^n]} as n 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.