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[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.
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See Also: order of vanishing

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

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Cross-references: irreducible polynomial, quotient, multiple, properties, number, open set, continuous, function, complex functions, complex number, divisible, polynomial
There are 29 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 5610 times total.

Classification:
AMS MSC: 12D10 (Field theory and polynomials :: Real and complex fields :: Polynomials: location of zeros )
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