Hilbert series

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.

These notions have been extended to filtered algebras, and graded or filtered modules over these algebras, as well as to coherent sheaves over projective schemes.

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.

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 ${\displaystyle S_{0}=K}$.

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