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I have a set of polynomials $P_t(z)= z^n+ a_{n-1}(t)z^{n-1}+\cdots+ a_0(t)$ which depends on a real parameter $t \in [a,b]$ and where $a_{n-1}(t),\ldots, a_0(t)$ are real continuous functions.

May I say that there exists a continuous map $\theta(t)$ such that $\theta(t)$ is a root of $P_t$ (for all $t$)?

I mean, I know that there exists a continuous dependence of the roots of a polynomial with respect to the coefficients and that the Viète map descends to a homeomorphism $w:C^n/S_n\to C^n$, but, can I 'choose' a root? Or I need the axiom of choice to affirm that there exists a map $C^n/S_n\to C^n$? In that case, may I get a such map to be continuous?

Any bibliography reference for all this?

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    Finding a function is easy, even without the AoC: any ordering of the set of complex numbers will be a well-ordering on the roots of $P_t(z)$ for each $t$. Choose the smallest root. Of course, this need not be continuous.2012-06-08
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    I have to wonder, why are you asking?2012-06-08

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Great question.

If the roots are all always real the answer is yes (this comes from the fact that in $\mathbb R$ you can order the roots from the lowest to the highest).

If the roots or the coefficients may be complex, the answer is in general negative. Take for example the polynomial $t^2-z \in \mathbb{C}[t]$, with $z \in \mathbb C$.

However there is a deep theorem (by Kato) that may help you: it states that if the roots of your polynomial depend only on a real parameter $t \in \mathbb {R}$ than you have $N$ continuous functions that describe the roots.

Anyway, I suggest you to give a look to Kato, Perturbation theory for linear operators, Springer (Theorem 5.2, pag 109 in my edition).

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    $\theta(t) = t^2$ is a continuous root of $t^2 - z$. If you meant $t - z^2$, then $\theta(t) = \begin{cases} \sqrt{t} & t \geq 0 \\ i \sqrt{-t} & t \leq 0 \end{cases}$ is.2012-06-08
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    You cannot define a continuous function $\omega: \mathbb C \to \mathbb C$ such that $\omega^2(z)=z$ for every $z \in \mathbb C$.2012-06-08
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    But you can define a continuous function on $[a,b] \subseteq \mathbb{R}$.2012-06-08
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    Yes, I agree with you, but I meant $z \in \mathbb C$ (I should have written that, sorry).2012-06-08
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    But does Kato's theorem use the axiom of choice? This is a key issue to the question.2012-06-08
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    @AsafKaragila: no, it doesn't (if I'm not wrong).2012-06-08
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    Thanks so much!!! A theorem as Kato's one is (I think) just what I need (maybe my question was no so clear) : My polynomial (of degree n) depends on a real parameter t, and I need to affirm that there are n continuous functions that describe (for each t) the n roots of my polynomial... Great suggestion! thanks again! :))2012-06-08
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    @Javier: you are welcome. ;-)2012-06-08