This work was motivated in part by the following general question: given an ideal I in a Cohen–Macaulay (abbreviated to CM) local ring R such that dim R/I=0, what information about I and its associated graded ring can be obtained from the Hilbert function and Hilbert polynomial of I? By the Hilbert (or Hilbert–Samuel) function of I, we mean the function HI(n) =λ(R/In) for all n[ges ]1, where λ denotes length. Samuel [23] showed that for large values of n, the function HI(n) coincides with a polynomial PI(n) of degree d=dim R. This polynomial is referred to as the Hilbert, or Hilbert–Samuel, polynomial of I. The Hilbert polynomial is often written in the form

formula here

where e0(I), [ctdot ], ed(I) are integers uniquely determined by I. These integers are known as the Hilbert coefficients of I.