2018/04/26 by Chalebgwa, Taboka
#Complex Variables (math.CV) #FOS: Mathematics
paper · doi:10.48550/arxiv.1804.09953
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