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Let $(R,\mathfrak m)$ be an analytically unramified local ring of positiveprime characteristic $p.$ For an ideal $I$, let $I^*$ denote its tight closure.We introduce the tight Hilbert function $H^*_I(n)=\ell(R/(I^n)^*)$ and thecorresponding tight Hilbert polynomial $P_I^*(n)$ where $I$ is an $\mathfrakm$-primary ideal. It is proved that $F$-rationality can be detected by thevanishing of the first coefficient of $P_I^*(n).$ We find the tight Hilbertpolynomial of certain parameter ideals in hypersurface rings andStanley-Reisner rings of simplicial complexes.

Tight Closure of powers of ideals and tight Hilbert polynomials