This talk will be based on McMullen’s paper “Families of rational maps and iterative root-finding algorithms”. A rigidity theorem on stable families of rational maps parameterized by complex manifolds is proven, showing that any such family is either affine or each member is conjugate by a Moebius transformation. This result is then applied to the case with the complex manifold is the space of monic, square-free polynomials to conclude that there does not exist any generally convergent purely iterative algorithm for polynomials of degree at least 4. |