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Roots of polynomial sequences in root-sparse regions

Christian Henriksen, Carsten Lunde Petersen, Eva Uhre

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 $μ_k$ of $q_k$ converge weak* to some compactly supported measure $μ$, 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 $p_μ$ 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.

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 iterates of a polynomial outside the filled-in Julia set. We also apply our result in the setting of extremal polynomials.
Paper Structure (7 sections, 10 theorems, 54 equations, 1 figure)

This paper contains 7 sections, 10 theorems, 54 equations, 1 figure.

Table of Contents

  1. Introduction and main results
  2. Prerequisites in potential theory
  3. Applications
  4. Families of orthogonal polynomials.
  5. Extremal families in the sense of Widom.
  6. Proofs of theorems
  7. Acknowledgments

Key Result

Theorem 1

Suppose that $(\mu_k)_k$ converges weak* to a compactly supported measure $\mu$. Let $U\subset \mathbb C$ be a root-sparse open set on which $p_\mu$ is nowhere locally constant. Then

Figures (1)

  • Figure 1: We illustrate the dynamics of $P(z)=z^2+\frac{1}{2}$ together with the roots of the second derivative of $P^k$, for $k = 2, 4$ (first row) and $k = 6, 10$ (second row). Every picture corresponds to the region $\{x +iy : -3/2 < x < 3/2, -3/2 < y < 3/2\}$. The filled-in Julia set $K(P)$ is shown in black. It is well-known that it is a Cantor set for this particular polynomial. The critical points of the Green's function $g_\Omega$ is shown in green, whereas the value of $g_\Omega$ is suggested by shades of red. Finally, the roots of the second derivative of $P^k$ are marked with blue crosses. A consequence of Proposition \ref{['pro:dyn']} is that there will be two roots of $\frac{dP^k}{dz^2}$ converging to each critical point of $g_\Omega$ as $k\to \infty$. Looking at the critical point at the origin, it does seem that two critical points of $g_\Omega$, lying on the imaginary axis, get closer and closer to $0$ as $k$ is increased.

Theorems & Definitions (21)

  • Definition 1
  • Theorem 1
  • Corollary 1
  • Corollary 2
  • Theorem 2
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • ...and 11 more