PlanetMath (more info)
 Math for the people, by the people.
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
sections
Encyclopædia
Papers
Books
Expositions

meta
Requests (193)
Orphanage
Unclass'd
Unproven (498)
Corrections (23)
Classification

talkback
Polls
Forums
Feedback
Bug Reports

downloads
Snapshots
PM Book

information
News
Docs
Wiki
ChangeLog
TODO List
Legalese
About
Owner confidence rating: Very high Entry average rating: Very high
[parent] multiplicity (Definition)

If a polynomial $ f(x)$ in $ \mathbb{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 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) \neq 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 derivative $ f'(x)$ with multiplicity $ m-1$.
  2. The zeros of the polynomial $ \gcd(f(x), f'(x))$ are same as the multiple zeros of $ f(x)$.
  3. The quotient $ \displaystyle\frac{f(x)}{\gcd(f(x), f'(x))}$ has the same zeros as $ f(x)$ but they all are simple.
  4. The zeros of any irreducible polynomial are simple.



"multiplicity" is owned by pahio.
(view preamble)

View style:

See Also: order of vanishing, derivative of polynomial

Other names:  order of the zero
Also defines:  zero of order, multiple zero, simple zero

This object's parent.
Log in to rate this entry.
(view current ratings)

Cross-references: irreducible polynomial, quotient, multiple, properties, number, open set, continuous, function, complex functions, zero of the polynomial, complex number, divisible, polynomial
There are 49 references to this entry.

This is version 10 of multiplicity, born on 2004-06-09, modified 2008-02-28.
Object id is 5905, canonical name is Multiplicity.
Accessed 5906 times total.

Classification:
AMS MSC: 12D10 (Field theory and polynomials :: Real and complex fields :: Polynomials: location of zeros )
Pending Errata and Addenda
None.
[ View all 2 ]
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | add derivation | add example | add (any)