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?