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Multiplicity

This article is about the mathematical term; Multiplicity is also the title of a 1996 film.

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In mathematics, multiplicity is a general term referring to 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 real or complex number a is called a root of multiplicity k of a polynomial p if there exists a polynomial s with:

<math>s(a) \neq 0<math>

and

p(x) = (x − a)^ks(x).

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

Example

The following polynomial p:

p(x) = x³ + 2x² − 7x + 4

has 1 and −4 as roots, and can be written as:

p(x) = (x + 4)(x − 1)²

This means that x = 1 is a root of multiplicity 2, and x = −4 is a 'simple' root (multiplicity 1).

In complex analysis

Let <math>z_0<math> be a root of a function f, and let n be the least positive integer m such that, the m-th derivative of f evaluated in <math>z = z_0<math> differs from zero:

<math>f^{(m)}(z_0)\neq0.<math>

Then the power series of <math>f<math> about <math>z_0<math> begins with the <math>n<math>th term, and <math>f<math> is said to have a root of multiplicity (or "order") <math>n<math>. If <math>n = 1<math>, the root is called a simple root (Krantz 1999, p. 70).

See also

• Fundamental theorem of algebra