Newton's method on Bring-Jerrard polynomials

Abstract

In this paper we study the topology of the hyperbolic component of the parameter plane for the Newton's method applied to $n$-degree Bring--Jerrard polynomials given by $P_n(z) = z^n-cz +1$, $c\in\mathbb{C}$. For $n=5$, using the Tschirnhaus--Bring--Jerrard nonlinear transformations, this family controls, at least theoretically, the roots of all quintic polynomials. We also study a bifurcation cascade of the bifurcation locus by considering $c\in \mathbb{R}$.