Roots of polynomial sequences in root‐sparse regions

Abstract

Given a family of polynomials, we call an open set root‐sparse if the number of zeros of is locally uniformly bounded on . We study the interplay between the individual zeros of the polynomials and those of the th derivatives in a root‐sparse open set , as . More precisely, if the root distributions of converge weak* to some compactly supported measure , whose potential is nowhere locally constant on a root‐sparse open set , then we link the roots of the th derivative , for an arbitrary , to the roots of and the critical points of the potential on compact subsets of . We apply this result in a polynomial dynamics setting to obtain convergence results for the roots of the th derivative of the iterates of a polynomial outside the filled‐in Julia set. We also apply our result in the setting of extremal polynomials.