It is well known that any quadratic polynomial is conjugated to a polynomial of the form $z^2+c$. How about polynomials with higher degree, such as $z^n+a_1z^{n-1}+\cdots+a_{n-1}$, is it also conjugated to a polynomial of the form $z^n+c$.
2026-03-26 17:51:42.1774547502
polynomial conjugate
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No. I assume you mean affinely conjugated, ie conjugated by means of affine map $h(z)=az+b$. Then you can see that you only have 2 parameters ($a$ and $b$) to simplify the form of the polynomial; there is no reason to expect that just 2 parameters will enable you to kill all but 1 coefficient.
What is true is that any polynomial $a_n z^n+ a_{n-1} z^{n-1}+\ldots$ is affinely conjugated to a monic centered polynomial, ie of the form $z^{n} + a_{n-2} z^{n-2} + \ldots$.