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Let $a_0, \ldots, a_n \in \mathbb{C}$, with $a_n \neq 0$. Consider set $$U_R = \{~z \in \mathbb{C} ~:~ |a_nz^n + \dots + a_1z + a_0| < R~\}$$ for each $R > 0$. How do I prove that $U_R$ is homeomorphic to a disk, if $R$ is large enough?

It's easy to see that for some small values of $R$, this isn't even connected, unless all roots of $P(z) = a_nz^n + \dots + a_0$ coincide. Is there a way to give a lower bound (depending on $a_n, \ldots, a_0$, of course) on the set of those $R$ for which $U_R$ is connected?

Thanks in advance.

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