Sunday, 1 November 2015

Hilbert series and Hilbert polynomial

Hilbert series and Hilbert polynomial

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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.
These notions have been extended to filtered algebras and graded filtered modules over these algebras.
The typical situations where these notions are used are the following:
The Hilbert series of an algebra or a module is a special case of the Hilbert–Poincaré series of a graded vector space.
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.

Contents

Definitions and main properties

Let us consider a finitely generated graded commutative algebra S over a field K, which is finitely generated by elements of positive degree. This means that
S = \bigoplus_{i \ge 0} S_i\
and that S_0=K.
The Hilbert function
HF_S\;:\;n\mapsto \dim_K\,S_n
maps the integer n onto the dimension of the K-vector space Sn. The Hilbert series, which is called Hilbert–Poincaré series in the more general setting of graded vector spaces, is the formal series
HS_S(t)=\sum_{n=0}^{\infty} HF_S(n)\,t^n.
If S is generated by h homogeneous elements of positive degrees d_1, \ldots, d_h, then the sum of the Hilbert series is a rational fraction
HS_S(t)=\frac{Q(t)}{\prod_{i=1}^h (1-t^{d_i})}\,,
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
HS_S(t)=\frac{P(t)}{(1-t)^\delta}\,,
where P is a polynomial with positive integer coefficients.
In this case the series expansion of this rational fraction is
HS_S(t)=P(t)\,\left(1+\delta\,t+\cdots +\binom{n+\delta-1}{\delta-1}\,t^n+\cdots\right)
where the binomial coefficient \binom{n+\delta-1}{\delta-1} is \;\frac{(n+\delta-1)(n+\delta-2)\cdots n}{(\delta-1)!}\; for n>-\delta and 0 otherwise.
This shows that there exists a unique polynomial HP_S(n) with rational coefficients which is equal to HF_S(n) for n\ge \deg P-\delta+1. This polynomial is the Hilbert polynomial. The least n0 such that HP_S(n)=HF_S(n) for n ≥ n0 is called the Hilbert regularity. It may be lower than \deg P-\delta+1.
The Hilbert polynomial is a numerical polynomial, since the dimensions are integers, but the polynomial almost never has integer coefficients (Schenck 2003, pp. 41).
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.
The Hilbert polynomial of a projective variety V in Pn is defined as the Hilbert polynomial of the homogeneous coordinate ring of V.

Graded algebra and polynomial rings

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 R_n=K[X_1,\ldots, X_n] onto S. Its kernel is a homogeneous ideal I and this defines an isomorphism of graded algebra between R_n/I 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.

Properties of Hilbert series

Additivity

Hilbert series and Hilbert polynomial are additive relatively to exact sequences. More precisely, if
0 \;\rightarrow\; A\;\rightarrow\; B\;\rightarrow\; C \;\rightarrow\; 0
is an exact sequence of graded or filtered modules, then we have
HS_B=HS_A+HS_C
and
HP_B=HP_A+HP_C.
This follows immediately from the same property for the dimension of vector spaces.

Quotient by a non-zero divisor

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
HS_{A/(f)}(t)=(1-t^d)\,HS_A(t)\,.
It follows immediately from the additivity on the exact sequence
0 \;\rightarrow\; A^{[d]}\; \xrightarrow{f}\; A \;\rightarrow\; A/f\rightarrow\; 0\,,
where the arrow labeled f is the multiplication by f and A^{[d]} 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 HS_{A^{[d]}}(t)=t^d\,HS_A(t)\,.

Hilbert series and Hilbert polynomial of a polynomial ring

The Hilbert series of the polynomial ring R_n=K[x_1, \ldots, x_n] in n indeterminates is
HS_{R_n}(t) = \frac{1}{(1-t)^{n}}\,.
It follows that the Hilbert polynomial is
HP_{R_n}(k) = {{k+n-1}\choose{n-1}} = \frac{(k+1)\cdots(k+n-1)}{(n-1)!}\,.
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 x_n) and remarking that HS_K(t)=1\,.

Shape of the Hilbert series and dimension

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 HS_{A/(f)}(t)=(1-t)\,HS_A(t). 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
HS_A(t)=\frac{P(t)}{(1-t)^d}
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!.

Degree of a projective variety and Bézout's theorem

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 I\subset k[x_0, x_1, \ldots, x_n], where k is a field, and let R=k[x_1, \ldots, x_n]/I 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 d hyperplanes in general position. This implies that the equations of these hyperplanes, say h_1, \ldots, h_{d}, are a regular sequence, and that we have the exact sequences
0 \;\rightarrow\; \left(R/\langle h_1,\ldots, h_{k-1}\rangle \right)^{[1]}\; \xrightarrow{h_k}\; R/\langle h_1,\ldots, h_{k-1}\rangle \;\rightarrow\; R/\langle h_1,\ldots, h_{k}\rangle\;\rightarrow\; 0,
for k=1, \ldots, d. This implies that
HS_{R/\langle h_1,\ldots, h_{d}\rangle}(t) = (1-t)^d\,HS_R(t)
is a polynomial, which is equal to the numerator P(t) of the Hilbert series of R. After dehomogenizing by putting x_0=1, Jordan-Hölder theorem for Artinian rings allows to prove that P(1) is the degree of the algebraic set V.
Similarly, if f is a homogeneous polynomial of degree \delta, which is not a zero divisor in R, the exact sequence
0 \;\rightarrow\; R^{[\delta]}\; \xrightarrow{f}\; R \;\rightarrow\; R/\langle f\rangle\;\rightarrow\; 0,
shows that
HS_{R/\langle f \rangle}(t)=(1-t^\delta)HS_R(t).
Looking on the numerators this proves the following generalization of Bézout's theorem:
If f is a homogeneous polynomial of degree \delta, 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 \delta .
The usual Bézout's theorem is easily deduced by starting from a hypersurface and intersecting it, one after the other, with n-1 other hypersurfaces.

Computation of Hilbert series and Hilbert polynomial

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, R=K[x_1,\ldots,x_n] 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.
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