For [Sendov's conjecture](https://en.wikipedia.org/wiki/Sendov%27s_conjecture), the distance 1 appears in the conjecture is tight, if one consider the polynomials $f\_{n}(z) = z^{n} - 1$ for all $n\geq 2$. I wonder if this polynomial is the local optima for the conjecture. More precisely, I want to know if the following statement is true:



> 
> For all $n\geq 2$, there exists $\epsilon = \epsilon\_{n} > 0$ such that for all degree $n$ polynomials $f(z) \in \mathbb{C}[z]$ with $\| f - f\_{n}\| < \epsilon$, $f$ satisfies the Sendov's conjecture.
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Here the norm is given as the $L^\infty$-norm of the coefficient vector ($\|a\_{n}z^{n} + \cdots + a\_{1}z + a\_{0}\| = \max\_{0\leq i\leq n} \{|a\_{i}|\}$). I couldn't find any works on Sendov's conjecture in this direction.



Local optimum for Sendov's conjecture
This follows from the work of


*Miller, Michael J.*, [**On Sendov’s conjecture for roots near the unit circle**](http://dx.doi.org/10.1006/jmaa.1993.1194), J. Math. Anal. Appl. 175, No. 2, 632-639 (1993). [ZBL0782.30007](https://zbmath.org/?q=an:0782.30007).


and independently


*Vâjâitu, Viorel; Zaharescu, A.*, [**Ilyeff’s conjecture on a corona**](http://dx.doi.org/10.1112/blms/25.1.49), Bull. Lond. Math. Soc. 25, No. 1, 49-54 (1993). [ZBL0796.30004](https://zbmath.org/?q=an:0796.30004).


who established Sendov's conjecture when the distinguished zero is sufficiently close to the unit circle. By Rouche's theorem, any sufficiently small perturbation of $f\_n$ will have its zeroes close enough to the unit circle for one of these two results to apply.


If one uses the more recent result of


*Kasmalkar, Indraneel G.*, [**On the Sendov conjecture for a root close to the unit circle**](http://ajmaa.org/cgi-bin/paper.pl?string=v11n1/V11I1P4.tex), Aust. J. Math. Anal. Appl. 11, No. 1, Article No. 4, 34 p. (2014). [ZBL1293.30018](https://zbmath.org/?q=an:1293.30018).


then one can obtain an explicit value of $\epsilon\_n$ for your question, probably of polynomial type in $n$ (although the asymptotic behavior in $n$ is not so relevant now, due to [my recent result](https://arxiv.org/abs/2012.04125) establishing the conjecture for all sufficiently large $n$).


