# Hilbert-Samuel polynomial

## Definition

Let be a Noetherian local ring with *unique* maximal ideal . Let be a finitely generated -module, and be an ideal of finite colength in (in other words, there exists a such that ).

Consider an essentially -adic filtration of , i.e. a descending chain:

such that there exists such that for .

The **Hilbert-Samuel function** for this filtration is a function sending a positive integer to the *length* of the quotient as an -module. This Hilbert-Samuel function turns out to be equal to a polynomial for sufficiently large values of . That polynomial is termed the Hilbert-Samuel polynomial for the filtration.

Some people use the term Hilbert-Samuel polynomial for the length polynomial, which is the polynomial measuring the length of the module .

When we simply talk of the Hilbert-Samuel polynomial, we by default refer to that for the ring as a module over itself. The default ideal is taken to be the maximal ideal and the default filtration is taken to be the standard one.

## Facts

- The specific Hilbert-Samuel polynomial depends on the specific filtration we choose. However, the
*leading coefficient*of the Hilbert-Samuel polynomial depends only on the ideal . - The leading coefficient depends on the specific choice of the Hilbert-Samuel polynomial. However, the degree is independent of the choice of .
- For a Noetherian local ring , the degree of the Hilbert-Samuel polynomial equals the Krull dimension.