Given a finitely generated 
-graded module 
over a graded ring 
(finitely generated over 
, which is an Artinian local ring), define the Hilbert
function of 
as the map 
such that, for all 
,
 |
(1)
|
where 
denotes the length. If 
is the dimension of 
,
then there exists a polynomial 
of degree 
with rational coefficients (called the Hilbert
polynomial of 
)
such that 
for all sufficiently large 
.
The power series
 |
(2)
|
is called the Hilbert series of 
. It is a rational function
that can be written in a unique way in the form
 |
(3)
|
where 
is a finite linear combination with integer coefficients of powers of 
and 
. If 
is positively graded, i.e., 
for all 
, then 
is an ordinary polynomial with integer coefficients in
the variable 
.
If moreover 
,
then 
,
i.e., the Hilbert series is a polynomial.
