multiplicity

If a polynomial $f(x)$ in $ℂ[x]$ is divisible by ${(x-a)}^m$ but not by ${(x-a)}^{m+1}$ ($a$ is some complex number,  $m\in {\mathbb{Z}}_{+}$), we say that   $x=a$   is a  zero of the polynomial with multiplicity $m$ or alternatively a zero of order $m$.

Generalization of the multiplicity to real (http://planetmath.org/RealFunction) and complex functions (by rspuzio):  If the function $f$ is continuous on some open set $D$ and  $f(a)=0$  for some  $a\in D$, then the zero of $f$ at $a$ is said to be of multiplicity $m$ if $\frac{f(z)}{{(z-a)}^m}$ is continuous in $D$ but $\frac{f(z)}{{(z-a)}^{m+1}}$ is not.

If  $m\geqq 2$, we speak of a multiple zero; if  $m=1$, we speak of a simple zero.  If  $m=0$, then actually the number $a$ is not a zero of $f(x)$, i.e.  $f(a)\ne 0$.

Some properties (from which 2, 3 and 4 concern only polynomials):

1. The zero $a$ of a polynomial $f(x)$ with multiplicity $m$ is a zero of the $f^{\prime }(x)$ with multiplicity $m-1$.

2. The zeros of the polynomial $\gcd (f(x),f^{\prime }(x))$ are same as the multiple zeros of $f(x)$.

3. The quotient ${\displaystyle \frac{f(x)}{\gcd (f(x),f^{\prime }(x))}}$ has the same zeros as $f(x)$ but they all are .

4. The zeros of any irreducible polynomial are .

Title	multiplicity
Canonical name	Multiplicity
Date of creation	2013-03-22 14:24:18
Last modified on	2013-03-22 14:24:18
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	14
Author	pahio (2872)
Entry type	Definition
Classification	msc 12D10
Synonym	order of the zero
Related topic	OrderOfVanishing
Related topic	DerivativeOfPolynomial
Defines	zero of order
Defines	multiple zero
Defines	simple zero
Defines	simple