Hilbert Series and Hilbert Polynomial - Definitions and Main Properties

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

and that .

The Hilbert function

maps the integer n onto the dimension of the K-vector space S_n. The Hilbert series, which is called Hilbert–Poincaré series in the more general setting of graded vector spaces, is the formal series

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 .

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 finetely 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.