Hilbert series and Hilbert polynomial

In <a href="/facts/Commutative_algebra/mKtUdPCT">commutative algebra</a>, the Hilbert function, the Hilbert polynomial, and the Hilbert series of a <a href="/facts/Graded_algebra/AIUjkSaP">graded commutative algebra</a> finitely generated over a <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a> 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 <a href="/facts/Filtered_algebra/jL7AG71G">filtered algebras</a>, and graded or filtered <a href="/facts/Module_(mathematics)/jP0LIhFL">modules</a> over these algebras, as well as to <a href="/facts/Coherent_sheaves/h2NoUN7e">coherent sheaves</a> over <a href="/facts/Projective_scheme/aB2XdB7B">projective schemes</a>.
The typical situations where these notions are used are the following:

<ul><li>The quotient by a homogeneous <a href="/facts/Ideal_(ring_theory)/vCvytduX">ideal</a> of a <a href="/facts/Multivariate_polynomial/Lzak8VVx">multivariate polynomial</a> <a href="/facts/Polynomial_ring/ua3vk8Ih">ring</a>, graded by the total degree.</li>
<li>The quotient by an ideal of a multivariate polynomial ring, filtered by the total degree.</li>
<li>The filtration of a <a href="/facts/Local_ring/gRTdg5Qx">local ring</a> by the powers of its <a href="/facts/Maximal_ideal/nf4DnKVS">maximal ideal</a>. In this case the Hilbert polynomial is called the <a href="/facts/Hilbert%E2%80%93Samuel_function/nnA4xmRG">Hilbert–Samuel polynomial</a>.</li></ul>
The <a href="/facts/David_Hilbert/1vhlHn4b">Hilbert</a> series of an algebra or a module is a special case of the <a href="/facts/Hilbert%E2%80%93Poincar%C3%A9_series/cVJBU8yR">Hilbert–Poincaré series</a> of a <a href="/facts/Graded_vector_space/Ff1maywV">graded vector space</a>.
The Hilbert polynomial and Hilbert series are important in computational <a href="/facts/Algebraic_geometry/rh7RHVp3">algebraic geometry</a>, 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 <a href="/facts/Hilbert_scheme/kJcbts4e">Hilbert scheme</a> and <a href="/facts/Quot_scheme/6Fs27uHM">Quot scheme</a>.

Hilbert series and Hilbert polynomial open-in-new

Hilbert series and Hilbert polynomial