Hilbert series and Hilbert polynomial

The Hilbert - Samuel polynomial is a term used in the mathematical sciences in commutative algebra and algebraic geometry. It is used there in the dimension theory and in the calculation of the intersection points. While the degree is important for the dimensional analysis, the coefficients for the theory of algebraic -sectional geometry to play a role. It was named after David Hilbert and Pierre Samuel.

This article deals with commutative algebra. In particular, all rings considered are commutative and have an identity element. Ring homomorphisms form elements from one to one elements. For more details see Commutative Algebra.

Definitions

It should be

A graduated ring with the following properties:

The function is

Called the Hilbert - Samuel function

Under the conditions of the definition ( and with these names ), the following applies:

• For large the Hilbert - Samuel function of a polynomial. It is and the highest coefficient of is positive.

This means that there is a: and a are such that applies to all:

This polynomial is called the Hilbert - Samuel polynomial

Dimension theory

Is a local ring with maximal ideal, and

The graduated ring to this ideal. Then for the degree of Hilbertpolynoms this ring ( considered as a module over itself):

( Is the Krull dimension of the ring )