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Let $\{a_i(x):\mathbb{R}\rightarrow \mathbb{C}\}$ be continuous functions, does there exist some continuous functions $\{\lambda_i(x)\}$ such that $$a_{n-1}(x) y^n+a_{n-2}(x) y^{n-1}+\cdots +a_1(x)y^2+y=(\lambda_{n-1}(x)y+1)(\lambda_{n-2}(x)y+1)\cdots(\lambda_1(x)y+1)y?$$

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Yes, the map from the coefficients of a polynomial to the (unordered multi-)set of its (complex) roots is continuous.

This is easy to prove if all roots are simple, then the root curves are as smooth as the coefficient curves, but it also holds in points with non-trivial root multiplicity, where the root curves are only Hölder continuous with a fractional Hölder exponent.

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  • $\begingroup$ can you give me a complete proof?Thx $\endgroup$
    – user92646
    Mar 28, 2014 at 1:24

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