The Hilbert-Kunz Function for Binomial Hypersurfaces

I give an iterative closed form formula for the Hilbert-Kunz function for any binomial hypersurface in general, over any field of arbitrary positive characteristic. I prove that the Hilbert-Kunz multiplicity associated with any binomial hypersurface over any field of arbitrary positive characteristic is rational. As an example, I also prove the well known fact that for 1-dimensional binomial hypersurfaces the Hilbert-Kunz multiplicity is a positive integer and give a precise account of the integer. 1. Introduction Let be a Noetherian local ring of dimension and of prime characteristic . Let be an -primary ideal and where is an integer. The “Hilbert-Kunz function” of with respect to is defined as where th Frobenious power of , that is, the ideal generated by , . The associated Hilbert-Kunz multiplicity is defined to be Monsky showed in his paper [1] that the limit exists and is a real constant. Further he showed that Several authors have investigated . They showed that is a rational number for certain special kind of binomial hypersurfaces [2], cubic curves and surfaces [3] and full flag varieties, and elliptic curves [4]. A binomial hypersurface is defined in the beginning of Section 4. In this paper, we are interested in giving an iterative closed form formula for the Hilbert-Kunz function for any binomial hypersurface in general. Our methods work for any positive characteristic. Our work generalizes the work of Conca [2]. In [2], Conca computes the Hilbert-Kunz function of monomial ideals and also of those Binomial hypersurfaces whose terms defining the hypersurface are relatively prime. In [2], Conca also proves that the Hilbert-Kunz multiplicity associated with these special Binomial Hypersurfaces is always rational. In this paper, we prove that the Hilbert-Kunz multiplicity associated with any binomial hypersurface over any field of arbitrary positive characteristic is rational. The rationality result appears in Eto’s work (Theorem 2.2, [5]), where the multiplicity is interpreted as a volume of a polytope. Here we give a different proof for the rationality result. Our work also generalizes the work [6] of Watanabe, where he deals only with normal toric varieties. The organization of this work is more or less clear from the table of contents. But to be precise, until Section 4 begins, the matter of this work holds true for any hypersurface over any field of positive characteristic and need not have to be a binomial hypersurface! Also, the filtration introduced in Section 2.1 is effective for any general ideal of a polynomial ring where is a field of