On unmixed and equi-dimensional associated graded rings

Let (A, m) be an analytically un-ramified Noetherian local ring of dimension d 1, I a regular m-primary ideal of A and let I be integral closure ideal of I. If A is of characteristic p > 0 then let I^* denote the tight closure of I. Let GI (A) =₍ ₀Iⁿ/I^n+1 be the associated graded ring of A with respect to I. Assume GI (A) is unmixed and equi-dimensional. We show that either the function P₈: \, n (Iⁿ/Iⁿ) is a polynomial type of degree d-1 or Iⁿ=Iⁿ for all n 1. We prove an analogus result for the tight closure filtration if A is of characteristic p > 0. When A is generalized Cohen-Macaulay and I is generated by standard system of parameters we give bounds for the first Hilbert coefficients of the integral closure filtration of I and the tight closure filtration of I.