Given a finitely generated -graded module M over a graded ring R (finitely generated over , which is an Artinian local ring), the Hilbert function of M is the map
such that, for all ,

where denotes the length. If n is the dimension of M, then there exists a polynomial of degree n with rational coefficients (called the Hilbert polynomialof M) such that for all sufficiently large a.
The power series

is called the Hilbert series of M. It is a rational function that can be written in a unique way in the form

where is a finite linear combination with integer coefficients of powers of t and .If M is positively graded, i.e., for all a < 0, then is an ordinary polynomial with integer coefficients in the variable t. If moreover , then , i.e., the Hilbert series is a polynomial.
If M has a finite graded free resolution

then

Moreover, if is a regular sequence over M of homogeneous elements of degree 1, then the Hilbert function of the -dimensional quotient module is

and in particular,

These properties suggest effective methods for computing the Hilbert series of a finitely generated graded module over the polynomial ring , where K is a field.
The Hilbert series of R, which has dimension n, can be obtained by considering the maximal regular sequence of R, and the Hilbert function of the 0-dimensional quotient ring , which is the same as K. Now , and for all . Hence . It follows that is the constant polynomial 1, so that

This approach can be applied to all Cohen-Macaulay quotient rings , where I is an ideal generated by homogeneous polynomials. The first step is to find a maximal regular sequence of S 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 S) whose Hilbert series is the polynomial . By (5) and (6) the result is

If, for example, , which is a 1-dimensional Cohen-Macaulay ring, an Artinian reduction is . Its Hilbert series can be easily determined from the definition: for all a < 0, whereas, for all , , since the length of a vector space over K is the same as its dimension. Since in all multiples of and are zero, we have



Hence, This is By (8) it follows that

The same result can be obtained by first constructing a graded free resolution of S over R,

which yields , whereas the remaining are zero. Hence, by (4) and (7),

as above. We rewrite it in form of a power series,

From this, according to (2), we can retrieve the values of the Hilbert function H(S,a),

It follows that the Hilbert polynomial of S is the constant polynomial
More generally, the graded free resolution of , where I is the ideal of , and f is a polynomial of degree d > 0, is

and the Hilbert series of S is

For more complicated ideals I, the computation requires the use of Gr&ouml;bner bases, with the techniques explained by Eisenbud (1995), Fr&ouml;berg (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 1889, Eisenbud et al. 1996). Let be a collection of m distinct points. Then the number of conditions imposed by on forms of degree d is called the Hilbert function or .If curves and of degrees d and e meet in a collection of points, then for any k , the number of conditions imposed by on forms of degree k is independent of and and is given by

where the binomial coefficient is taken as 0 if a < 2 (Cayley 1843).