Multiplicity (mathematics)

In mathematics, multiplicity is a general term meaning "the number of values for which a given condition holds." For example, the term is used to refer to the value of the totient valence function, or the number of times a given polynomial equation has a root at a given point.

Multiplicity of a root of a polynomial

A number $a\in \mathbb {R}$ is called a root of multiplicity k of a polynomial p if there exists a polynomial s with:

$s(a)\neq 0$

and

p(x) = (x - a)^ks(x).

If k equals 1, then a is a simple root.

Example

The following polynomial p:

p(x) = x³ - x² - x + 1

has 1 and −1 as roots, and can be written as:

p(x) = (x + 1)(x - 1)²

This means that x = 1 is a root of multiplicity 2, and x = −1 is a 'normal' root (multiplicity 1).

See also Fundamental Theorem of Algebra, separable polynomial, intersection number.

In complex analysis

Let $n_{0}$ be a root of a function f, and let n be the least positive integer m such that

$f^{(M)}(z_{0})\neq 0$.

Then the power series of $f$ about $z_{0}$ begins with the $n$th term, and $f$ is said to have a root of multiplicity (or "order") $n$. If $n=1$, the root is called a simple root (Krantz 1999, p. 70).

External link

"Multiplicity" on MathWorld