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.