Tight closure of powers of ideals and tight hilbert polynomials.

Let (R,) be an analytically unramified local ring of positive prime characteristic p. For an ideal I, let I* denote its tight closure. We introduce the tight Hilbert function HI*(n) = ℓ(R/In)) and the corresponding tight Hilbert polynomial PI*(n), where I is an m-primary ideal. It is proved that F-rationality can be detected by the vanishing of the first coefficient of PI*(n). We find the tight Hilbert polynomial of certain parameter ideals in hypersurface rings and Stanley-Reisner rings of simplicial complexes. [ABSTRACT FROM AUTHOR]