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

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.