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 quotient by a homogeneous ideal of a multivariate polynomial ring, graded by the total degree. The quotient by an ideal of a multivariate polynomial ring, filtered by the total degree. The filtration of a local ring by the powers of its maximal ideal. In this case the Hilbert polynomial is called the Hilbert–Samuel polynomial. The Hilbert series of an algebra or a module is a special case of the Hilbert–Poincaré series of a graded vector space. The 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. In addition, they provide useful invariants for families of algebraic varieties because a flat family π : X → S {\displaystyle \pi :X\to S} has the same Hilbert polynomial over any closed point s ∈ S {\displaystyle s\in S} . This is used in the construction of the Hilbert scheme and Quot scheme.
Definitions and main properties 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 = ⨁ i ≥ 0 S i {\displaystyle S=\bigoplus _{i\geq 0}S_{i}}
and that S 0 = K {\displaystyle S_{0}=K} . The Hilbert function
H F S : n ⟼ dim K S n {\displaystyle HF_{S}:n\longmapsto \dim _{K}S_{n}}
maps the integer n to 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
H S S ( t ) = ∑ n = 0 ∞ H F S ( n ) t n . {\displaystyle 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 , … , d h {\displaystyle d_{1},\ldots ,d_{h}} , then the sum of the Hilbert series is a rational fraction
H S S ( t ) = Q ( t ) ∏ i = 1 h ( 1 − t d i ) , {\displaystyle HS_{S}(t)={\frac {Q(t)}{\prod _{i=1}^{h}\left(1-t^{d_{i}}\right)}},}
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
H S S ( t ) = P ( t ) ( 1 − t ) δ , {\displaystyle HS_{S}(t)={\frac {P(t)}{(1-t)^{\delta }}},}
where P is a polynomial with integer coefficients, and δ {\displaystyle \delta } is the Krull dimension of S. In this case the series expansion of this rational fraction is
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