Multiplicities

Multiplicities describe how many times a given value or feature occurs in a mathematical object, typically as a root, eigenvalue, or zero of a function. They quantify the strength or order of that occurrence and play a key role in factorization, spectral theory, and analysis.

In polynomials, a is a root of multiplicity m if (x − a) raised to the power m divides the polynomial, but (x − a)^(m+1) does not. Equivalently, p(x) can be written as (x − a)^m q(x) with q(a) ≠ 0. A simple root has multiplicity 1, and the sum of all root multiplicities equals the degree of the polynomial.

In linear algebra, eigenvalues have algebraic and geometric multiplicities. The algebraic multiplicity of an eigenvalue λ is its multiplicity as a root of the characteristic polynomial det(A − λI) = 0. The geometric multiplicity is the dimension of the eigenspace corresponding to λ. For any matrix, the geometric multiplicity is between 1 and the algebraic multiplicity, and the sum of all algebraic multiplicities equals the size of the matrix.

In analysis, for a holomorphic function f, the order of vanishing (the multiplicity) at a point z0 is the largest m such that f(z) has a zero of order m at z0, i.e., f(z) = (z − z0)^m g(z) with g(z0) ≠ 0. The same notion applies to zeros and poles of meromorphic functions in complex geometry and can be extended via divisor theory to record these on curves.

Multiplicities are nonnegative integers; higher multiplicity indicates a stronger or more tangential interaction with the value or direction under consideration. They influence factorization, perturbation behavior, and counting arguments across algebra and analysis.