Hilbert-Samuel polynomial

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

This article is about commutative algebra. In particular, all rings under consideration are commutative and have a one element. Ring homomorphisms map single elements onto single elements. For more details, see Commutative Algebra .


Be it

a graduated ring with the following properties:

  1. is a module of finite length
  2. is created as a ring of and finitely many elements .
  3. be a graduated finite module.

Then the function

Called the Hilbert-Samuel function

Under the conditions of the definition (and with these designations) the following sentence applies:

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

This means that there is a: and a , so that applies to all :

This polynomial is called the Hilbert-Samuel polynomial

Dimensional theory

Is a local ring with maximum ideal , and

the graduated ring to this ideal. Then the following applies to the degree of the Hilbert polynomial of this ring (considered as a module over itself):

( is the Krull dimension of the ring)