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Sendov’s Conjecture: A Note on a Paper of Dégot
- Source :
- Analysis Mathematica. 46:447-463
- Publication Year :
- 2020
- Publisher :
- Springer Science and Business Media LLC, 2020.
-
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, Degot 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 Degot’s paper.
Details
- ISSN :
- 1588273X and 01333852
- Volume :
- 46
- Database :
- OpenAIRE
- Journal :
- Analysis Mathematica
- Accession number :
- edsair.doi...........2724545fa2616ae69bd5df4b8336ab09