abstract: One of the classical problems in all areas of mathematics is to find roots of complex polynomials. It is well known that this can be done only by methods of approximation. We discuss three classical methods: the Newton, Weierstrass, and Ehrlich-Aberth methods; these are complex analytic maps that, under iteration, are supposed to converge to one root, resp. all roots of the polynomial. Locally, these methods converge fast, but the global dynamical properties are hard to describe.
We introduce these complex analytic dynamical systems and describe recent progress towards their global dynamical properties. In particular, 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, but there are many open questions for all these methods.
Much of this is joint work with members of my ERC team, in particular my PhD student Bernhard Reinke, as well as with colleagues.