Tight closure of powers of parameter ideals in hypersurface rings and their tight Hilbert polynomials

TL;DR

This paper investigates the tight closure of powers of parameter ideals in certain diagonal hypersurface rings, revealing Cohen-Macaulay properties and explicitly determining tight Hilbert polynomials in specific cases.

Contribution

It provides explicit descriptions of tight closures and tight Hilbert polynomials for diagonal hypersurface rings under various conditions, advancing understanding of their algebraic structure.

Findings

Tight closure of parameter ideals computed for specific hypersurfaces.

Associated graded rings are Cohen-Macaulay in many cases.

Explicit tight Hilbert polynomials derived for particular hypersurfaces.

Abstract

In this paper we find the tight closure of powers of parameter ideals of certain diagonal hypersurface rings. In many cases the associated graded ring with respect to tight closure filtration turns out to be Cohen-Macaulay. This helps us find the tight Hilbert polynomial in these diagonal hypersurfaces. We determine the tight Hilbert polynomial in the following cases: (1) F-pure diagonal hypersurfaces where number of variables is equal to the degree of defining equation, (2) diagonal hypersurface rings where characteristic of the ring is one less than the degree of defining equation and (3) quartic diagonal hypersurface in four variables.