A quadratic approximation to the Sendov radius near the unit circle

Abstract:

Define $S(n,\beta )$ to be the set of complex polynomials of degree $n\ge 2$ with all roots in the unit disk and at least one root at $\beta$. For a polynomial $P$, define $|P|_\beta$ to be the distance between $\beta$ and the closest root of the derivative $P’$. Finally, define $r_n(\beta )=\sup \{ |P|_\beta : P \in S(n,\beta ) \}$. In this notation, a conjecture of Bl. Sendov claims that $r_n(\beta )\le 1$. In this paper we investigate Sendov’s conjecture near the unit circle, by computing constants $C_1$ and $C_2$ (depending only on $n$) such that $r_n(\beta )\sim 1+C_1(1-|\beta |)+C_2(1-|\beta |)^2$ for $|\beta |$ near $1$. We also consider some consequences of this approximation, including a hint of where one might look for a counterexample to Sendov’s conjecture.