Multiplicité

Multiplicité is a term used across disciplines to denote the number or degree of repetition of a particular object or property. In mathematical contexts, it most often refers to how many times a value, root, or feature occurs within a given structure.

In algebra and analysis, multiplicité describes the order of a root of a polynomial. If a polynomial p(x) has a root r such that (x − r)^m divides p(x) but (x − r)^(m+1) does not, then r has multiplicity m. This is the multiplicity of the root. In complex analysis, a similar idea applies to zeros and poles of analytic functions, where the order (multiplicity) indicates how strongly the function vanishes or blows up at a point.

In linear algebra, the term also appears as algebraic multiplicity and geometric multiplicity of an eigenvalue. The algebraic multiplicity of an eigenvalue λ is its multiplicity as a root of the characteristic polynomial, while the geometric multiplicity is the dimension of the corresponding eigenspace. These notions help describe the behavior of matrices and linear transformations.

In geometry and topology, multiplicity can describe how objects intersect or coincide, such as the intersection multiplicity of curves at a point. In multiset theory and combinatorics, multiplicity refers to the number of occurrences of an element within a multiset.

Beyond pure mathematics, multiplicity appears in statistics and data analysis when considering repeated measurements or multiple comparisons, and in various applied fields to express redundancy or the prevalence of certain features.

See also: multiplicity (disambiguation), root multiplicity, eigenvalue multiplicity, intersection theory.