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Just a thought that came up-- consider a polynomial $$P(y)=a_ni^n(x+iy)^n+a_{n-1}i^{n-1}(x+iy)^{n-1}+...+a_0.$$

Is there a way of finding some value of $y$ for which the polynomial is nonzero for any given $x$? I think a closed form expression might be possible, but after some bashing with the binomial theorem I have $$P(y)=\sum_{m=0}^n\left(a_mi^m\left(\sum_{k=0}^m \frac{m!}{(m-k)!k!}x^k(iy)^{m-k}\right)\right),$$ and solving for $y$ here doesn't seem too fun. Perhaps someone would know a better way to go about this? Thanks in advance.

Edit: I'm thinking this is identical to solving a polynomial of order $n$, in which case explicit solutions for the roots are impossible if $n>4$.

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    Are you assuming the $x,y$ to be real or complex? If it is the latter, there is no real (no pun intended) point in writing $x+iy$: for any fixed $x$, the value of $x+iy$ could be any complex number, and one can easily convert between $y$ and $x+iy$ and back.2012-03-22
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    Sorry, should have been more explicit: $x,y\in \mathbb{R}$.2012-03-22

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