Roots of polynomial sequences in root-sparse regions

Given a family $(q_k)_k$ of polynomials, we call an open set $U$ root-sparse if the number of zeros of $q_k$ is locally uniformly bounded on $U$. We study the interplay between the individual zeros of the polynomials $q_k$ and those of the $m$th derivatives $q_k^{(m)}$, in a root-sparse open set $U$, as $k\to\infty$. More precisely, if the root distributions $\mu_k$ of $q_k$ converge weak* to some compactly supported measure $\mu$, whose potential is nowhere locally constant on a root-sparse open set $U$, then we link the roots of the $m$th derivative $q_k^{m}$, for an arbitrary $m>0$, to the roots of $q_k$ and the critical points of the potential $p_\mu$ on compact subsets of $U$. We apply this result in a polynomial dynamics setting to obtain convergence results for the roots of the $m$th derivative of iterates of a polynomial outside the filled-in Julia set. We also apply our result in the setting of extremal polynomials.