In commutative algebra the Hilbert–Samuel function, named after David Hilbert and Pierre Samuel, of a nonzero finitely generated module M over a commutative Noetherian local ring A and a primary ideal I of A is the map such that, for all , where \ell denotes the length over A. It is related to the Hilbert function of the associated graded module by the identity For sufficiently large n, it coincides with a polynomial function of degree equal to, often called the Hilbert-Samuel polynomial (or Hilbert polynomial).

Examples

For the ring of formal power series in two variables kx,y taken as a module over itself and the ideal I generated by the monomials x2 and y3 we have

Degree bounds

Unlike the Hilbert function, the Hilbert–Samuel function is not additive on an exact sequence. However, it is still reasonably close to being additive, as a consequence of the Artin–Rees lemma. We denote by P_{I, M} the Hilbert-Samuel polynomial; i.e., it coincides with the Hilbert–Samuel function for large integers. Proof: Tensoring the given exact sequence with R/I^n and computing the kernel we get the exact sequence: which gives us: The third term on the right can be estimated by Artin-Rees. Indeed, by the lemma, for large n and some k, Thus, This gives the desired degree bound.

Multiplicity

If A is a local ring of Krull dimension d, with m-primary ideal I, its Hilbert polynomial has leading term of the form for some integer e. This integer e is called the multiplicity of the ideal I. When I=m is the maximal ideal of A, one also says e is the multiplicity of the local ring A. The multiplicity of a point x of a scheme X is defined to be the multiplicity of the corresponding local ring.