Given a finitely generated 
-graded module 
over a graded ring 
(finitely generated over 
, which is an Artinian local ring), the Hilbert function of 
is 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.
If 
has a finite graded free resolution
 |
(4)
|
then
 |
(5)
|
Moreover, if 
is a regular sequence over 
of homogeneous elements of degree 1, then the Hilbert function
of the 
-dimensional
quotient module 
is
 |
(6)
|
and in particular,
 |
(7)
|
These properties suggest effective methods for computing the Hilbert series of a finitely generated graded module over the polynomial ring ![R=K[X_1,...,X_n]](/images/equations/HilbertFunction/Inline32.svg)
, where 
is a field.
The Hilbert series of 
,
which has dimension 
,
can be obtained by considering the maximal regular sequence 
of 
, and the Hilbert function of the 0-dimensional quotient
ring 
,
which is the same as 
.
Now 
,
and 
for all 
.
Hence 
.
It follows that 
is the constant polynomial 1, so that
 |
(8)
|
This approach can be applied to all Cohen-Macaulay quotient rings 
, where 
is an ideal generated by homogeneous polynomials. The first
step is to find a maximal regular sequence 
of 
composed of homogeneous polynomials of degree 1; here, by
virtue of the Cohen-Macaulay property, 
. This will produce a 0-dimensional ring 
(a so-called Artinian reduction
of 
)
whose Hilbert series is the polynomial 
. By (5) and (6) the result is
 |
(9)
|
If, for example, ![S=K[X_1,X_2]/<X_1^2>](/images/equations/HilbertFunction/Inline53.svg)
,
which is a 1-dimensional Cohen-Macaulay ring,
an Artinian reduction is ![S^_=S/<X_2>=K[X_1,X_2]/<X_1^2,X_2>](/images/equations/HilbertFunction/Inline54.svg)
. Its Hilbert
series can be easily determined from the definition: 
for all 
, whereas, for all 
, 
, since the length of a vector space over
is the same as its dimension. Since
in 
all multiples of 
and 
are zero, we have
 |  |  |
(10)
|
 |  |  |
(11)
|
 |  |  |
(12)
|
Hence, 
This is 
.
By (8) it follows that
 |
(13)
|
The same result can be obtained by first constructing a graded free resolution of 
over 
,
 |
(14)
|
which yields 
,
whereas the remaining 
are zero. Hence, by (4) and (7),
 |  |  |
(15)
|
 |  |  |
(16)
|
 |  |  |
(17)
|
as above. We rewrite it in the form of a power series,
 |  |  |
(18)
|
 |  |  |
(19)
|
From this, according to (2), we can retrieve the values of the Hilbert function 
,
 |
(20)
|
It follows that the Hilbert polynomial of 
is the constant polynomial 
.
More generally, the graded free resolution of 
, where 
is the ideal 
of ![R=K[X_1,...,X_n]](/images/equations/HilbertFunction/Inline99.svg)
, and 
is a polynomial of degree 
, is
 |
(21)
|
and the Hilbert series of 
is
 |
(22)
|
For more complicated ideals 
, the computation requires the use of Gröbner
bases, with the techniques explained by Eisenbud (1995), Fröberg (1997),
or Kreuzer and Robbiano (2000).
Historically, the Hilbert function arises in algebraic geometry for the study of finite sets of points in the projective
plane as follows (Cayley 1843, Eisenbud et al. 1996). Let 
be a collection of 
distinct points. Then the number of conditions imposed by
on forms of degree 
is called the Hilbert function 
of 
. If curves 
and 
of degrees 
and 
meet in a collection 
of 
points, then for any 
, the number 
of conditions imposed by 
on forms of degree 
is independent of 
and 
and is given by
 |
(23)
|
where the binomial coefficient 
is taken as 0 if 
(Cayley 1843).
