To find all roots of a given complex polynomial in one variable is one of the classical problems in all areas of mathematics. It is well known that this can be done only by methods of approximation. We discuss three classical methods: the Newton, Weierstrass (=Durand-Kerner), and Ehrlich-Aberth methods; these are complex analytic dynamical systems that are supposed to converge to one root, resp. all roots of the polynomial. These methods converge fast locally, but the global dynamical properties are a big challenge. 

We will describe these three methods and their relations and differences. We have worked on them for a long time. In this talk we will focus on recent progress on all these methods from the dynamical systems point of view, partly with help of computer algebra —where we reached the current computational limits. We also describing current challenges.

Several of the results we describe have been obtained in recent years with my ERC team here in Marseille, in particular with Bernhard Reinke. The Newton and Weierstrass methods are not globally convergent: for open sets of polynomials there are open sets of initial points that fail to converge to roots. Moreover, for Weierstrass and Ehrlich-Aberth, there are orbits that are always defined and converge, but not to roots. For Newton, there is meanwhile a rich theory about its global dynamics, including reasonable complexity bounds, but there are many open questions for all these methods.