In commutative algebra, the Hilbert function, the Hilbert polynomial, and the Hilbert series of a graded commutative algebra finitely generated over a field are three strongly related notions which measure the growth of the dimension of the homogeneous components of the algebra.
The Hilbert polynomial and Hilbert series are important in computational algebraic geometry, as they are the easiest known way for computing the dimension and the degree of an algebraic variety defined by explicit polynomial equations.
The Hilbert function maps the integer n to the dimension of the K-vector space Sn.
, then the sum of the Hilbert series is a rational fraction where Q is a polynomial with integer coefficients.
If S is generated by elements of degree 1 then the sum of the Hilbert series may be rewritten as where P is a polynomial with integer coefficients, and
is the Krull dimension of S. In this case the series expansion of this rational fraction is where is the binomial coefficient for
All these definitions may be extended to finitely generated graded modules over S, with the only difference that a factor tm appears in the Hilbert series, where m is the minimal degree of the generators of the module, which may be negative.
Conversely, if S is a graded algebra generated over the field K by n homogeneous elements g1, ..., gn of degree 1, then the map which sends Xi onto gi defines an homomorphism of graded rings from
onto S. Its kernel is a homogeneous ideal I and this defines an isomorphism of graded algebra between
and S. Thus, the graded algebras generated by elements of degree 1 are exactly, up to an isomorphism, the quotients of polynomial rings by homogeneous ideals.
Therefore, the remainder of this article will be restricted to the quotients of polynomial rings by ideals.
More precisely, if is an exact sequence of graded or filtered modules, then we have and This follows immediately from the same property for the dimension of vector spaces.
Let A be a graded algebra and f a homogeneous element of degree d in A which is not a zero divisor.
Then we have It follows from the additivity on the exact sequence where the arrow labeled f is the multiplication by f, and
indeterminates is It follows that the Hilbert polynomial is The proof that the Hilbert series has this simple form is obtained by applying recursively the previous formula for the quotient by a non zero divisor (here
In this section, one does not need irreducibility of algebraic sets nor primality of ideals.
has Krull dimension one, and is the ring of regular functions of a projective algebraic set
is an Artinian ring, which is a k-vector space of dimension P(1), and Jordan–Hölder theorem may be used for proving that P(1) is the degree of the algebraic set V. In fact, the multiplicity of a point is the number of occurrences of the corresponding maximal ideal in a composition series.
, which is not a zero divisor in R, the exact sequence shows that Looking on the numerators this proves the following generalization of Bézout's theorem: In a more geometrical form, this may restated as: The usual Bézout's theorem is easily deduced by starting from a hypersurface, and intersecting it with n − 1 other hypersurfaces, one after the other.
A projective algebraic set is a complete intersection if its defining ideal is generated by a regular sequence.
In fact, these formulas imply that, if a graded free module L has a basis of h homogeneous elements of degrees
This section describes how the Hilbert series may be computed in the case of a quotient of a polynomial ring, filtered or graded by the total degree.
Finally let B be a Gröbner basis of I for a monomial ordering refining the total degree partial ordering and G the (homogeneous) ideal generated by the leading monomials of the elements of B.
In fact the Gröbner basis may be computed by linear algebra over the polynomials of degree bounded by the regularity.
For example in both Maple and Magma these functions are named HilbertSeries and HilbertPolynomial.
In algebraic geometry, graded rings generated by elements of degree 1 produce projective schemes by Proj construction while finitely generated graded modules correspond to coherent sheaves.
is a coherent sheaf over a projective scheme X, we define the Hilbert polynomial of
The Euler characteristic in this case is a well-defined number by Grothendieck's finiteness theorem.
the associated coherent sheaf the two definitions of Hilbert polynomial agree.
is equivalent to the category of graded-modules modulo a finite number of graded-pieces, we can use the results in the previous section to construct Hilbert polynomials of coherent sheaves.