The Hilbert function of a finitely generated module 
, graded over a standard graded
ring 
that is Noetherian and whose degree-zero part
is an Artinian local ring, is the map 
such that, for all 
,
 |
(1)
|
where 
denotes module length. If 
is nonzero of Krull
dimension 
, its eventual Hilbert
polynomial 
has degree 
and rational coefficients,
with 
for sufficiently large 
. For a module of Krull
dimension 0, the eventual Hilbert polynomial
is zero.
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 the Krull dimension of 
and 
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 ring regular sequence over 
of homogeneous elements of degree
1, then the Hilbert series 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/Inline31.svg)
, where 
is a field.
The Hilbert series of 
, which has Krull
dimension 
, can be obtained by considering the maximal ring
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)
|
Over an infinite field, this approach can be applied to Cohen-Macaulay quotient rings 
, where 
is an ideal generated by homogeneous
polynomials. The first step is to find a maximal ring
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 
. The result is
 |
(9)
|
If, for example, ![S=K[X_1,X_2]/<X_1^2>](/images/equations/HilbertFunction/Inline52.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/Inline53.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 
. 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,
 |  |  |
(15)
|
 |  |  |
(16)
|
 |  |  |
(17)
|
as above. We rewrite it in the form of a power series,
 |  |  |
(18)
|
 |  |  |
(19)
|
From this expansion, 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/Inline98.svg)
, and 
is a homogeneous polynomial
of total 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).
