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 Ī 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)=n≥ 0Iⁿ/Iⁿ⁺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 PĪ :\,n↦ λ(Iⁿ̄/Iⁿ) is a polynomial type of degree $d-1$ or Iⁿ̄=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.