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.
- 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.