Suppose $p$ is a polynomial of degree $n$ and all roots $z_1,\cdots,z_n $ of $p$ are inside the unit disk. Then how to show that every disk of radius $\sqrt{2}$ and centered at $z_k$ for $k=1,\cdots,n$ contains a critical point.
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$\begingroup$ What makes you pick $\sqrt{2}$ here? $\endgroup$– Yemon ChoiDec 2, 2014 at 19:37
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1$\begingroup$ This feels to be strongly connected with Gauss-Lucas, en.wikipedia.org/wiki/Gauss%E2%80%93Lucas_theorem $\endgroup$– Per AlexanderssonDec 2, 2014 at 20:11
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4$\begingroup$ @PerAlexandersson AFAIK work on Sendov's conjecture is motivated by the observation that one seems to always be able to do better than Gauss-Lucas, but I don't see why G-L is more relevant to the weak version here than to the original Sendov conjecture $\endgroup$– Yemon ChoiDec 2, 2014 at 23:32
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$\begingroup$ As a matter of fact there is way smaller bounds for the radius for instance one is $2^{1/n}$ which can be find here But I am trying to come up with a simplified proof. $\endgroup$– BigMDec 2, 2014 at 23:42
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$\begingroup$ @BigM Can you give a link that works? I am interested in this question. $\endgroup$– Steven GubkinDec 3, 2014 at 19:45
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