A function determined by a hypersurface of positive characteristic

Let R be a formal power series ring over a perfect field k of prime characteristic p, and let m be the maximal ideal of R. Suppose f is a non-zero element in m. In this paper, we introduce a function xi (x) associated with a hypersurface defined on the closed interval [0, 1] in R. The Hilbert-Kunz function and the F-signature of a hypersurface appear as the values of our function xi (x) on the interval s endpoints.