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

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