Multiplicidades

Multiplicidades is a term used across mathematics and related disciplines to denote the order, size, or number of occurrences of a given phenomenon. Although the exact meaning depends on the context, the common idea is to quantify how strongly a feature appears or how many times it occurs with a certain structure.

In algebra, the multiplicity of a root of a polynomial is the exponent of its corresponding factor. If f(x) = (x − a)^m g(x) with g(a) ≠ 0, then a is a root of multiplicity m. In analysis and complex analysis, a zero of a holomorphic function at a point z0 has a multiplicity equal to the largest m such that f(z) = (z − z0)^m h(z) with h(z0) ≠ 0; poles have a dual notion, describing the order of divergence.

In algebraic geometry, multiplicity describes the order of contact or interaction at a point. The multiplicity of a point p on a plane curve defined by f is the smallest total degree of nonzero terms in the Taylor expansion of f about p. Intersection multiplicity measures how two curves meet at a point, accounting for tangency and higher-order contact; Bezout’s theorem relates the sum of these multiplicities to degrees of the curves.

In representation theory, multiplicities count how many times an irreducible representation occurs in a decomposition of a representation: a representation V may decompose as a direct sum of irreducibles with certain multiplicities.

In physics and chemistry, multiplicities (or degeneracies) describe the number of distinct states sharing the same energy level or quantum number.