Abstract
Sendov’s conjecture states that if all the zeroes of a complex polynomial P(z) of degree at least two lie in the unit disk, then within a unit distance of each zero lies a critical point of P(z). In a paper that appeared in 2014, Dégot proved that, for each a ∈ (0, 1), there exists an integer N such that for any polynomial P(z) with degree greater than N, if P(a) = 0 and all zeroes lie inside the unit disk, the disk |z − a| ≤ 1 contains a critical point of P(z). Based on this result, we derive an explicit formula N(a) for each a ∈ (0, 1) and, consequently obtain a uniform bound N for all a ∈ [α, β] where 0 < α < β < 1. This (partially) addresses the questions posed in Dégot’s paper.
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Acknowledgments
I would like to thank Gareth Boxall for many insightful comments and suggestions, and Florian Breuer for introducing me to this problem. I would also like to thank the referees for spotting several errors in the original manuscript as well as for their helpful comments which led to an improved presentation of the results in this paper. This work is based on the research supported in part by the National Research Foundation of South Africa (Grant Number 96234), and was completed under a sponsorship provided by the Fields Institute for Research in Mathematical Sciences.
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Chalebgwa, T.P. Sendov’s Conjecture: A Note on a Paper of Dégot. Anal Math 46, 447–463 (2020). https://doi.org/10.1007/s10476-020-0050-x
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DOI: https://doi.org/10.1007/s10476-020-0050-x