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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.
Research in the Mathematical Sciences – Springer Journals
Published: Dec 1, 2021
Keywords: Tight closure; Parameter ideals; Powers of ideals; Tight Hilbert polynomial; Diagonal hypersurface; Primary: 13A35; Secondary: 13D40
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