Geometry of Complex Polynomials: On Sendov's Conjecture

[摘要] ENGLISH ABSTRACT : Sendov's conjecture states that if all the zeroes of a complex polynomialP(z) of degree at least two lie in the unit disk, then within a unit distanceof each zero lies a critical point of P(z). In a paper that appeared in 2014,Dégot proved that, for each α ε (0, 1), there is an integer N such that for anypolynomial P(z) with degree greater than N, P(a) = 0 and all zeroes insidethe unit disk, the disk │z- α│ ≤ 1 contains a critical point of P(z). Basingon this result, we derive an explicit formula N(a) for each α ε (0, 1) and,furthermore, obtain a uniform bound N for all a ε [α,β] where 0 < α < β < 1. This addresses the questions posed in Dégot's paper.

[发布机构] Stellenbosch University