Polynomials with Zeros on the Unit Circle: Regularity of Leja Sequences

TL;DR

This paper studies a greedy polynomial sequence with roots on the unit circle, proving that roots become evenly distributed in angle at a rate of at most $( ext{log} N)^2 / N$, even with adversarial points.

Contribution

It introduces a new greedy construction of polynomials with roots on the unit circle and proves their roots' equidistribution rate, linking it to a dynamical system involving the inverse fractional Laplacian.

Findings

Roots of the polynomial sequence equidistribute in angle at rate at most $( ext{log} N)^2 / N$

Regularity persists even with adversarial point additions

Connection established between polynomial roots distribution and inverse fractional Laplacian dynamics

Abstract

Let $z_1, \dots, z_m$ be $m$ distinct complex numbers, normalized to $|z_k| = 1$, and consider the polynomial $$ p_{m}(z) = \prod_{k=1}^{m}{(z-z_k)}.$$ We define a sequence of polynomials in a greedy fashion, $$ p_{N+1}(z) = p_{N}(z) \left(z - z^*\right)\qquad \mbox{where}~z^* = \arg\max_{|z|=1} |p_{N}(z)|,$$ and prove that, independently of the initial polynomial $p_m$, the roots of $p_{N}$ equidistribute in angle at rate at most $(\log{N})^2/N$. This even persists when sometimes adding `adversarial' points by hand. We rephrase the main result in terms of a dynamical system involving the inverse fractional Laplacian $(-\Delta)^{-1/2}$ and conjecture that, when phrased in this language, the underlying regularity phenomenon might appear in a very general setting.