Let (R;m) be a Noetherian local ring and let I be an R-ideal. The associated graded ring of I, G = grI(R), plays a significant role in the study of resolution of singularities. Its relevance lies upon the fact that it represents algebraically the exceptional fiber of the blowup of a variety along a subvariety. A commonly addressed issue is to find numerical conditions that imply lower bounds on the depth of G . In [7, 8] and [3], for instance, this depth has been measured by using the Hilbert coefficients of I. To better explain these results, let us introduce some notation: An ideal J I is called a reduction of I if Ir+1 = JIr for some integer r. The least such r is called the reduction number of I with respect to J, and denoted rJ(I). If R is Cohen– Macaulay with infinite residue field and I is an m-primary ideal, then any minimal (with respect to inclusion) reduction of I is generated by a regular sequence. The Hilbert–Samuel function of I is the numerical function HI(n) = λ(R=In) (where λ( ) denotes length) that measures the growth of the length of R=In for all n 1. If d denotes the dimension of R, it is well-known that for n 0, HI(n) is a polynomial in n of degree d