Hilbert Series and Hilbert Polynomial

Hilbert Series And Hilbert Polynomial

Given a graded commutative algebra finitely generated over a field, the Hilbert function, the Hilbert series and the Hilbert polynomial are three strongly related notions which measure the growth of the dimension of its homogeneous components.

These notions have been extended to filtered algebras and graded of filtered modules over these algebras.

The typical situations where these notion 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.

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.

Read more about Hilbert Series And Hilbert Polynomial:  Definitions and Main Properties, Graded Algebra and Polynomial Rings, Computation of Hilbert Series and Hilbert Polynomial

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