Given a nonzero finitely generated module 
over a commutative Noetherian local ring 
with maximal ideal 
and a proper ideal 
of 
, the Hilbert-Samuel function of 
with respect to 
is the map 
such that, for all 
,
 |
where 
denotes the length. It is related to the Hilbert
function of the associated graded module 
by the identity
 |
For sufficiently large 
, it coincides with a polynomial function of degree equal to
.
