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The (probably) famous Gauss-Lucas theorem states that the roots of the derivative $P'(z)$ are contained in the convex hull of the roots of $P(z)$, where $P(z)$ is complex variable polynomial.

I am interested here in could it be the case that we always have some polynomial of any degree (except $1$) $P(z)$ such that some root of its derivative is "at a small as we want distance" from some root of $P(z)$.

To be more precise, here is the statement of the question:

Is it true that for every $\varepsilon>0$ and for every $n\in \mathbb N \setminus \{1\}$ there exists polynomial $P(z)$ in complex variable of degree $n$ with $n$ different roots such that there is root $z_a$ of $P'(z)$ and root $z_b$ of $P(z)$ which are such that we have $|P'(z_a)-P(z_b)|<\varepsilon$

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  • $\begingroup$ Yes, start with a polynomial with a double root and perturb the coefficients. $\endgroup$ – Michael Burr Nov 14 '15 at 1:11
  • $\begingroup$ @MichaelBurr All roots are different in the question, if that changes anything? $\endgroup$ – Farewell Nov 14 '15 at 1:13
  • $\begingroup$ If you insist that the coefficients of the polynomial are bounded integers, then the answer is no. $\endgroup$ – Michael Burr Nov 14 '15 at 1:13
  • $\begingroup$ No, it doesn't. After perturbation, the roots will all be different, but because the roots depend continuously on the coefficients, there will be a root of the derivative arbitrarily close to a root of the function. $\endgroup$ – Michael Burr Nov 14 '15 at 1:14
  • $\begingroup$ @MichaelBurr If we choose some $\varepsilon_0>0$ is there an easy way to construct some concrete polynomial for every degree $n>1$ (it is a sequence of polynomials for every concrete $\varepsilon>0$ ) such that the question holds? I do not see immediately that this is the case. $\endgroup$ – Farewell Nov 14 '15 at 1:20
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Let $P(z) = (z-\epsilon)(z-2\epsilon)\cdots (z-n\epsilon).$ Using the mean value theorem we can see that every root of $P'$ is less than $\epsilon$ away from some root of $P.$

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Let $P(z)=(z-\epsilon)(z+\epsilon)\prod_{i=1}^{n-2}(z-i)$. This is a perturbation of the polynomial $z^2\prod_{i=1}^{n-2}(z-i)$, removing the double root.

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