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What does the following mean?

The roots of a polynomial are a continuous function of the coefficients

Please guide me with an example.

S.C.B.
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    Roughly speaking, it means that "changing a little your polynomial gives you little changes of the roots". I think that this can be totally formalized, but it would be tedious and technical. – Crostul Apr 07 '16 at 06:51
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    Related: http://math.stackexchange.com/questions/63196/continuity-of-the-roots-of-a-polynomial-in-terms-of-its-coefficients – Henry Apr 07 '16 at 06:51

1 Answers1

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This means that given a root $x_0$ of a polynomial $P_0(X)= a_{0,0}+a_{1,0}X++a_{n,0}X^n$, there exits a continuous function $f$ of the coefficents $f(a_0,a_1,...a_n)$, defined in the neigbourhood of $(a_{0,0},...,a_{n,0})$ such that $f(a_0,a_1,...a_n)$ is a root of the polynomial $P(X)=a_{0}+a_{1}X++a_{n}X^n$.

Unfortunately, this theorem is false : let $P_0(X)= X^2$ : theres is no continuous function $r(a_0)$ defined in the neigbourhood of $a_{0,0}=0$ such that $r(a_0)^2+a_0=0$ (neither with real nor with complex coefficients)

However such a function exists in the neigbourhood of a simple root, as an immediate application of the implicit function theorem.

Thomas
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    The correct generalization (which works fine for multiple roots) is to take disks of radius $\epsilon>0$ (in $\Bbb C$) around all the roots of $P_0$. Then there is a $\delta>0$ so that if we change the coefficients by less than $\delta$, all the roots of the perturbed polynomial are contained in the union of those disks. – Ted Shifrin Apr 07 '16 at 17:26
  • Certainly, but the roots can also desapear ! In particular the map from the set of polynomials of degree n to the set of finite sets of $\bf C^d$ which send a polynomial to its set of root is not continuous for the Hausdorff topology – Thomas Apr 08 '16 at 04:36
  • Yes, but my statement is true nevertheless, working with complex polynomials. – Ted Shifrin Apr 08 '16 at 05:30
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    Thanks for the great example. I am still a bit confused here. Given that the roots vary continuously with the parameters https://www.ams.org/journals/proc/1965-016-01/S0002-9939-1965-0171902-8/S0002-9939-1965-0171902-8.pdf why can't I define continuous functions mapping coefficients to roots by using the disk definition given in the paper? – lychtalent May 01 '23 at 09:53