multiroot

Multiroot, in mathematics, is a term used for a root of a polynomial that occurs with multiplicity greater than one. It is more commonly referred to as a multiple root. If P(x) is a polynomial over a field F and r lies in an algebraic closure of F, then r is a multiroot of P with multiplicity m ≥ 2 when P(r) = 0 and (x − r) appears to the m-th power in the factorization of P.

A standard way to express this is P(x) = (x − r)^m Q(x) where Q(r) ≠ 0. In many settings, especially over fields of characteristic zero, a number r is a multiroot of P exactly when r is a common root of P and the derivative P'. Equivalently, gcd(P, P') ≠ 1 indicates the existence of a multiple root. The total degree of P equals the sum of the multiplicities of its roots, counting over the algebraic closure. The discriminant of P vanishes if and only if P has a multiple root.

Examples illustrate the concept: P(x) = x^2 has a double root at x = 0; P(x) = (x − 1)^3(x + 2) has a triple root at x = 1.

Detection and computation often rely on gcd(P, P') and factorization. Discriminants and resultants provide global criteria for the presence of multiroots. In applications, multiroots relate to tangency and singularities in algebraic geometry, influence numerical root-finding stability, and affect the structure of polynomial factorization in algorithms. See also multiplicity, polynomial, discriminant, and derivative.