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 $$H_I^*\left( n \right) = \Im \left( {R/\left( {{I^n}} \right)*} \right)$$ and the corresponding tight Hilbert polynomial $$P_I^*\left( n \right)$$, where I is an m-primary ideal. It is proved that F-rationality can be detected by the vanishing of the first coefficient of $$P_I^*\left( n \right)$$. We find the tight Hilbert polynomial of certain parameter ideals in hypersurface rings and Stanley-Reisner rings of simplicial complexes.