Abstract
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.
Highlights
Let (A, n) be a Noetherian local ring of dimension d and of prime characteristic p > 0
Where I[q] = nth Frobenious power of I, that is, the ideal generated by xq, x ∈ I
Monsky showed in his paper [1] that the limit exists and is a real constant
Summary
Let (A, n) be a Noetherian local ring of dimension d and of prime characteristic p > 0. We are interested in giving an iterative closed form formula for the Hilbert-Kunz function for any binomial hypersurface in general. In [2], Conca computes the Hilbert-Kunz function of monomial ideals and of those Binomial hypersurfaces whose terms defining the hypersurface are relatively prime. In [2], Conca proves that the Hilbert-Kunz multiplicity associated with these special Binomial Hypersurfaces is always rational. We prove that the Hilbert-Kunz multiplicity associated with any binomial hypersurface over any field of arbitrary positive characteristic is rational. Algebra fact that for any 1-dimensional binomial hypersurface, the associated Hilbert-Kunz multiplicity is a positive integer and I give a precise account of the integer.
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