On the distance to a root of polynomials (Q657130)

Summary: In [Ergodic Theory Dyn. Syst. 22, No. 3, 935--945 (2002; Zbl 1011.37024)], \textit{D. Schleicher} gave an explicit estimate of an upper bound for the number of iterations of Newton's method it takes to find all roots of polynomials with prescribed precision. In this paper, we provide a method to improve the upper bound given by Schleicher. We give here an iterative method for finding an upper bound for the distance between a fixed point \(z\) in an immediate basin of a root \(\alpha\) to \(\alpha\), which leads to a better upper bound for the number of iterations of Newton's method.