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Given a graded commutative algebra finitely generated over a field, the Hilbert function, Hilbert polynomial, and Hilbert series are three strongly related notions which measure the growth of the dimension of its homogeneous components.
The typical situations where these notions are used are the following:
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.
and that .
The Hilbert function
If S is generated by h homogeneous elements of positive degrees , 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 degrees 1 then the sum of the Hilbert series may be rewritten as
where P is a polynomial with positive integer coefficients.
In this case the series expansion of this rational fraction is
where the binomial coefficient is for and 0 otherwise.
This shows that there exists a unique polynomial with rational coefficients which is equal to for . This polynomial is the Hilbert polynomial. The least n0 such that for n ≥ n0 is called the Hilbert regularity. It may be lower than .
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.
The Hilbert function, the Hilbert series and the Hilbert polynomial of a filtered algebra are those of the associated graded algebra.
Polynomial rings and their quotients by homogeneous ideals are typical graded algebras. 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.
Hilbert series and Hilbert polynomial are additive relatively to exact sequences. More precisely, if
is an exact sequence of graded or filtered modules, then we have
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 immediately from the additivity on the exact sequence
where the arrow labeled f is the multiplication by f and is the graded algebra, which is obtained from A by shifting the degrees by d, in order that the multiplication by f has degree 0. This implies that
The Hilbert series of the polynomial ring 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 ) and remarking that
A graded algebra A generated by homogeneous elements of degree 1 has Krull dimension zero if the maximal homogeneous ideal, that is the ideal generated by the homogeneous elements of degree 1, is nilpotent. This implies that the dimension of A as a K-vector space is finite and the Hilbert series of A is a polynomial P(t) such that P(1) is equal to the dimension of A as a K vector space.
If the Krull dimension of A is positive, there is a homogeneous element f of degree one which is not a zero divisor (in fact almost all elements of degree one have this property). The Krull dimension of A/f is the Krull dimension of A minus one.
The additivity of Hilbert series shows that . Iterating this a number of times equal to the Krull dimension of A, we get eventually an algebra of dimension 0 whose Hilbert series is a polynomial P(t). This show that the Hilbert series of A is
where the polynomial P(t) is such that P(1) ≠ 0 and d is the Krull dimension of A.
This formula for the Hilbert series implies that the degree of the Hilbert polynomial is d and that its leading coefficient is P(1)/d!.
The Hilbert series allows us to compute the degree of an algebraic variety as the value at 1 of the numerator of the Hilbert series. This provides also a simple proof of Bézout's theorem. For this purpose, let us consider an projective algebraic set V defined as the set of the zeros of a homogeneous ideal , where k is a field, and let be the ring of the regular functions on the algebraic set (in this section, we do not need that the algebraic set be irreducible nor that the ideal is prime).
If the dimension of V, equal to the dimension of R is d, the degree of V is the number of points of intersection, counted with multiplicity, of V with the intersection of hyperplanes in general position. This implies that the equations of these hyperplanes, say are a regular sequence, and that we have the exact sequences
for This implies that
is a polynomial, which is equal to the numerator of the Hilbert series of R. After dehomogeneizing by putting , Jordan-Hölder theorem for Artinian rings allows to prove that is the degree of the algebraic set V.
Similarly, if f is a homogeneous polynomial of degree , which is not a zero divisor in R, the exact sequence
Looking on the numerators this proves the following generalization of Bézout's theorem.theorem:
If f is a homogeneous polynomial of degree , which is not a zero divisor in R, then the degree of the intersection of V with the hypersurface defined by f is the product of the degree of V by .
The usual Bézout's theorem is easily deduced by starting from a hypersurface and intersecting it, one after the other, with other hypersurfaces.
The Hilbert polynomial is easily deducible from the Hilbert series. 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.
Thus let K a field, be a polynomial ring and I be an ideal in R. Let H be the homogeneous ideal generated by the homogeneous parts of highest degree of the elements of I. If I is homogeneous, then H=I. 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.
The computation of the Hilbert series is based on the fact that the filtered algebra R/I and the graded algebras R/H and R/G have the same Hilbert series.
Thus the computation of the Hilbert series is reduced, through the computation of a Gröbner basis, to the same problem for an ideal generated by monomials, which is usually much easier than the computation of the Gröbner basis. The computational complexity of the whole computation depends mainly on the regularity, which is the degree of the numerator of the Hilbert series. In fact the Gröbner basis may be computed by linear algebra over the polynomials of degree bounded by the regularity.
The computation of Hilbert series and Hilbert polynomials are available in most computer algebra systems. For example in both Maple and Magma these functions are named HilbertSeries and HilbertPolynomial.