# Hilbert-Samuel polynomial

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 .

## Definitions

Be it

${\ displaystyle R = \ bigoplus _ {n \ in \ mathbb {N}} R_ {n}}$

a graduated ring with the following properties:

1. ${\ displaystyle R_ {0}}$is a module of finite length${\ displaystyle R_ {0}}$
2. ${\ displaystyle R}$is created as a ring of and finitely many elements .${\ displaystyle R_ {0}}$${\ displaystyle a_ {1}, \ dots, a_ {r}}$
3. ${\ displaystyle M}$be a graduated finite module.${\ displaystyle R}$

Then the function

${\ displaystyle f \ colon \ mathbb {N} \ to \ mathbb {N}}$
${\ displaystyle f \ colon n \ mapsto l_ {R_ {0}} \ left (\ bigoplus _ {i = 0} ^ {n-1} M_ {i} \ right)}$

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.${\ displaystyle n}$${\ displaystyle P [X]}$${\ displaystyle \ mathbb {Q} [X]}$${\ displaystyle grad (P [X]) \ leq r}$${\ displaystyle P [X]}$

This means that there is a: and a , so that applies to all : ${\ displaystyle P [X] \ in \ mathbb {Q} [X]}$${\ displaystyle: k \ in \ mathbb {N}}$${\ displaystyle n> k}$

${\ displaystyle f (n) = P (n)}$

This polynomial is called the Hilbert-Samuel polynomial

### Dimensional theory

Is a local ring with maximum ideal , and ${\ displaystyle A}$${\ displaystyle m}$

${\ displaystyle \ mathrm {gr} _ {m} (A): = \ bigoplus _ {n \ in \ mathbb {N}} m ^ {n} / m ^ {n + 1}}$

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): ${\ displaystyle \ mathrm {P} _ {m} (\ mathrm {gr} _ {m} (A))}$

${\ displaystyle \ mathrm {grad} (\ mathrm {P} _ {m} (\ mathrm {gr} _ {m} (A))) = \ mathrm {dim} (A)}$

( is the Krull dimension of the ring) ${\ displaystyle \ mathrm {dim} (A)}$