If G_I(A) is unmixed and equidimensional then either λ(Ī^n/I^n) is a polynomial of degree d-1 or Ī^n = I^n for all n; analogous result holds for tight closure, plus bounds on Hilbert coefficients in the generalized Cohen-Macaulay case.
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On unmixed and equi-dimensional associated graded rings
If G_I(A) is unmixed and equidimensional then either λ(Ī^n/I^n) is a polynomial of degree d-1 or Ī^n = I^n for all n; analogous result holds for tight closure, plus bounds on Hilbert coefficients in the generalized Cohen-Macaulay case.