Sendov's Conjecture: A Note on a Paper of Degot

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 distance of each zero lies a critical point ofP(z). In a paper that appeared in 2014, Degot proved that, for eacha is an element of (0, 1), there exists an integerNsuch that for any polynomialP(z) with degree greater thanN, ifP(a) = 0 and all zeroes lie inside the unit disk, the disk |z-a| <= 1 contains a critical point ofP(z). Based on this result, we derive an explicit formulaN(a) for eacha is an element of (0, 1) and, consequently obtain a uniform boundNfor alla is an element of [alpha, beta] where 0 < alpha < beta < 1. This (partially) addresses the questions posed in Degot's paper.