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A quantitative result on Sendov’s conjecture for a zero near the unit circle. (English) Zbl 1232.30008

Let \(p\) be a complex polynomial all of whose zeros lie in the closed unit disk. Concerning Sendov’s conjecture, it is known that if a zero \(a\) of \(p\) lies “sufficiently close” to the unit circle, then the distance from \(a\) to the closest critical point is less than 1. In a previous paper, the author obtained an upper bound for the radius of the disk, centered at the origin, which contains all the critical points of \(p\). In the paper under review, the author improves the aforementioned upper bound and then estimates the range of the zero \(a\) such that the distance from \(a\) to the closest critical point is less than 1. This result implies that if a zero of \(p\) (\(\deg(p)=n\)), is close to the unit circle and all the critical points are far from the zero, then the polynomial \(p\) must be close to \(z^n-c\) with \(|c|=1\).

MSC:

30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
26C10 Real polynomials: location of zeros
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