Multiplicity

A numerical invariant (often Hilbert–Samuel multiplicity) that measures the leading coefficient of a Hilbert polynomial or the asymptotic growth rate of lengths of M/I^nM for an m-primary ideal I in a local ring; it quantifies local algebraic concentration or singularity order.

Multiplicity is the integer count describing how many times a given root, eigenvalue, or intersection occurs: algebraic (root) multiplicity is the exponent of a factor (x−α)^m in a polynomial, and algebraic multiplicity of an eigenvalue is its multiplicity as a root of the characteristic polynomial; geometric multiplicity counts independent eigenvectors.