Dynamics of higher-dimensional root finding methods

I will give an overview about the dynamics of higher-dimensional iterative root finding methods: given a complex polynomial in one variable of some degree d, these methods are based on iteration in d complex variables and should converge to a vector consisting of the d roots. I will focus on the "Weierstrass-(Durand-Kerner)" and the "Ehrlich-Aberth" methods. A fundamental question is whether these methods are generally convergent (so whether they "always work" as observed in practice), an obstruction for this is the existence of attracting cycles. Together with Dierk Schleicher and Micheal Stoll we established the existence of attracting cycles for the Weierstrass method. While the question of general convergence for the Ehrlich-Aberth method is still open, I would like to present some connections of the search for attracting cycles with the enumerative geometry of moduli spaces of rational curves, based on joint work with Rob Silversmith.