Roots of polynomial sequences in root-sparse regions

TL;DR

This paper investigates the relationship between the zeros of polynomial sequences and their derivatives within root-sparse regions, revealing how root distributions and potential theory influence derivative roots, with applications in polynomial dynamics and extremal polynomials.

Contribution

It establishes a link between the roots of polynomial derivatives and the original roots and potentials in root-sparse regions, extending understanding in polynomial dynamics and extremal polynomial theory.

Findings

Roots of derivatives relate to original roots and potential critical points.

Convergence of root distributions influences derivative root locations.

Applications include polynomial dynamics and extremal polynomial analysis.

Abstract

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.