Following up on an answer I got on a previous question, here, given a polynomial $p(z) = \sum_{k=0}^n a_k z^k$ over $\mathbb{C}$, how do I construct (or prove existence of) a biholomorphic map $w=\varphi(z)$, such that $p(\varphi(z))=w^n.$ Thanks.
Biholomorphically transform a polynomial into a monomial
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complex-analysis
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0Well, the initial question was to find $\varphi$ s.t. $p(\varphi(z))=\varphi(z)^n$ where $n=deg(p)$. With my polynomial, it is certainly not possible. – 2012-09-04
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This isn't true globally, since a biholomorphism will preserve the number of distinct zeros of a given function, so if $p$ has more than one zero, there is no such biholomorphism.
However, I think the person leaving that comment meant that this is true locally: in a small neighborhood of a zero of $p$, there is a biholomorphism with a small neighborhood of $0$ such that $p$ goes to $w^m$, where $m$ is the order of the zero.