A Uniform Bound for the Distance to a Root of Complex Polynomials Under Newton’s Method

Let $N_p$ denote the Newton map induced by a complex polynomial $p$. In 2002, Schleicher showed that there exists a uniform bound $\mathcal{A}_d$ such that for every polynomial $p$ of degree $d$ and for every point $z_0$ in the immediate basin of a root $\alpha$ of $p$, we have $ | z_0-\alpha | \leq \mathcal{A}_d | N_p(z_0)-z_0 |$. Schleicher also presented that $\mathcal{A}_d\le f_d$, where $ f_d=\frac{d^2(d-1)}{2(2d-1)}\binom{2d}{d} \sim \frac{4^{d-1}d^2}{\sqrt{\pi d}}$. In 2020, the authors showed that $\mathcal{A}_d lt;\frac{3}{\sqrt{d}}(3.02)^d$ when $d\ge 12.$ The goal of this paper is to establish a better bound for $\mathcal{A}_d$ by using collaboration between roots of $p$. We establish that $\mathcal{A}_d lt;(1.77)^d$ for all sufficiently large $d$. As a consequence, it gives a better bound of the expected total number of iterations of Newton s method required to reach all roots of every polynomial $p$ within a given precision.