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Hilbert Function


Given a finitely generated Z-graded module M over a graded ring R (finitely generated over R_0, which is an Artinian local ring), the Hilbert function of M is the map H(M,i):Z->Z such that, for all a in Z,

 H(M,a)=l(M_a),
(1)

where l denotes the length. If n is the dimension of M, then there exists a polynomial P_M(x) of degree n with rational coefficients (called the Hilbert polynomial of M) such that P_M(a)=H(M,a) for all sufficiently large a.

The power series

 H_M(t)=sum_(a in Z)H(M,a)t^a
(2)

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

 H_M(t)=(Q_M(t))/((1-t)^d),
(3)

where Q_M(t) is a finite linear combination with integer coefficients of powers of t and t^(-1). If M is positively graded, i.e., M_a=0 for all a<0, then Q_M(t) is an ordinary polynomial with integer coefficients in the variable t. If moreover dim(M)=0, then H_M(t)=Q_M(t), i.e., the Hilbert series is a polynomial.

If M has a finite graded free resolution

 0-> direct sum _(j in Z)R(-j)^(beta_(sj))->...-> direct sum _(j in Z)R(-j)^(beta_(1j))-> direct sum _(j in Z)R(-j)^(beta_(0j))->M->0,
(4)

then

 H_M(t)=H_R(t)sum_(i,j)(-1)^ibeta_(ij)t^j.
(5)

Moreover, if x_1,...,x_r is a regular sequence over M of homogeneous elements of degree 1, then the Hilbert function of the n-r-dimensional quotient module M^_=M/<x_1,...,x_r>M is

 H_(M^_)(t)=(Q_M(t))/((1-t)^(n-r)),
(6)

and in particular,

 Q_(M^_)(t)=Q_M(t).
(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], where K is a field.

The Hilbert series of R, which has dimension n, can be obtained by considering the maximal regular sequence X_1,...,X_n of R, and the Hilbert function of the 0-dimensional quotient ring R^_=R/<X_1,...,X_n>, which is the same as K. Now H(K,0)=1, and H(K,a)=0 for all a!=0. Hence H_K(t)=1. It follows that Q_R(t) is the constant polynomial 1, so that

 H_R(t)=1/((1-t)^n).
(8)

This approach can be applied to all Cohen-Macaulay quotient rings S=R/I, where I is an ideal generated by homogeneous polynomials. The first step is to find a maximal regular sequence f_1,...f_m of S composed of homogeneous polynomials of degree 1; here, by virtue of the Cohen-Macaulay property, m=dim(S). This will produce a 0-dimensional ring S^_=S/<f_1,...f_m> (a so-called Artinian reduction of S) whose Hilbert series is the polynomial Q_(S^_). By (5) and (6) the result is

 H_S(t)=(Q_(S^_)(t))/((1-t)^m).
(9)

If, for example, S=K[X_1,X_2]/<X_1^2>, 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>. Its Hilbert series can be easily determined from the definition: H(S^_,a)=0 for all a<0, whereas, for all a>=0, H(S^_,a)=dim_K(S^__a), since the length of a vector space over K is the same as its dimension. Since in S^_ all multiples of X_1^2 and X_2 are zero, we have

S^__0=K
(10)
S^__1=KX_1
(11)
S^__a=0, for all a>=2.
(12)

Hence, H_(S^_)(t)=1+t. This is Q_(S^_)(t). By (8) it follows that

 H_S(t)=(1+t)/(1-t).
(13)

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

 0->R(-2)->^(1|->X_1^2)R->S->0,
(14)

which yields beta_(00)=beta_(12)=1, whereas the remaining beta_(ij) are zero. Hence, by (4) and (7),

H_S(t)=H_R(t)(1-t^2)
(15)
=(1-t^2)/((1-t)^2)
(16)
=(1+t)/(1-t),
(17)

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

H_S(t)=(1-t)/(1-t)+(2t)/(1-t)
(18)
=1+2t+2t^2+2t^3+...,
(19)

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

 {H(S,a)=0   for a<0; H(S,0)=1 ; H(S,a)=2   for a>=1.
(20)

It follows that the Hilbert polynomial of S is the constant polynomial P_S(x)=2.

More generally, the graded free resolution of S=R/I, where I is the ideal I=<f> of R=K[X_1,...,X_n], and f is a polynomial of degree d>0, is

 0->R(-d)->^(1|->f)R->S->0,
(21)

and the Hilbert series of S is

 H_S(t)=(1+t+t^2+...+t^(d-1))/((1-t)^(n-1)).
(22)

For more complicated ideals I, 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 Gamma={p_1,...,p_m} subset P^2 be a collection of m distinct points. Then the number of conditions imposed by Gamma on forms of degree d is called the Hilbert function h_Gamma of Gamma. If curves X_1 and X_2 of degrees d and e meet in a collection Gamma of d·e points, then for any k, the number h_Gamma(k) of conditions imposed by Gamma on forms of degree k is independent of X_1 and X_2 and is given by

 h_Gamma(k)=(k+2; 2)-(k-d+2; 2)-(k-e+2; 2)+(k-d-e+2; 2),
(23)

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


See also

Gröbner Basis, Hilbert Polynomial, Hilbert-Samuel Function, Hilbert Series, Module Multiplicity

Portions of this entry contributed by Margherita Barile

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References

Bruns, W. and Herzog, J. Cohen-Macaulay Rings, 2nd ed. Cambridge, England: Cambridge University Press, 1993.Cayley, A. "On the Intersection of Curves." Cambridge Math. J. 3, 211-213, 1843. Reprinted in Collected Math Papers I. Cambridge, England: Cambridge University Press, pp. 25-27, 1889.Eisenbud, D. Commutative Algebra with a View toward Algebraic Geometry. New York: Springer-Verlag, 1995.Eisenbud, D.; Green, M.; and Harris, J. "Cayley-Bacharach Theorems and Conjectures." Bull. Amer. Math. Soc. 33, 295-324, 1996.Fröberg, R. An Introduction to Gröbner Bases. Chichester, England: Wiley, 1997.Kreuzer, M. and Robbiano, L. Computational Commutative Algebra 1. Berlin: Springer-Verlag, 2000.Matsumura, H. Commutative Ring Theory. Cambridge, England: Cambridge University Press, 1986.

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Hilbert Function

Cite this as:

Barile, Margherita and Weisstein, Eric W. "Hilbert Function." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HilbertFunction.html

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