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 $$\underset{n\to \infty }{\lim }(\frac{1}{p^n}{)}^{\text{dim}(M)}\sum _{j=-\infty }^{\infty }\lambda \left((\frac{M}{I^{[p^n]}M}{)}_j\right)e^{-iyj/p^n}$$ exists. We prove that the limiting function in the complex variable $y$ is entire and name this function the \textit{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 $\frac{R}{I^{[p^n]}}$ 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.

v3: In Rmk 5.5, an error is fixed; a reference is added. Other minor changes