Solving the Sextic by Iteration: A Study in Complex Geometry and Dynamics

Recently, [Doyle and McMullen 1989] devised an iterative solution to the fifth degree polynomial. At the method’s core is a rational mapping f of CP with the icosahedral symmetry of a general quintic. Algebraically, this means that f commutes with a group of Möbius transformations that is isomorphic to the alternating groupA5. Moreover, thisA5-equivariant posseses nice dynamics: for almost any initial point a ∈ CP, the sequence of iterates fk(a) converges to one of the periodic cycles that comprise an icosahedral orbit.1 This breaking of A5-symmetry provides for a reliable or generally-convergent quintic-solving algorithm: with almost any fifth-degree equation, associate a rational mapping that has nice dynamics and whose attractor consists of a single orbit from which one computes a root. An algorithm that solves the sixth-degree equation calls for a dynamical system with S6 or A6 symmetry. Since neither S6 nor A6 acts on CP, attention turns to higher dimensions. Acting on CP is an A6-isomorphic group of projective transformations found by Valentiner in the late nineteenth century. The present work exploits this 2-dimensional A6 “soccer ball” in order to discover a “Valentiner-symmetric” rational mapping of CP whose dynamics experimentally appear to be nice in the above sense—transferred to the CP setting. This map provides the central feature of a conjecturally-reliable sextic-solving algorithm analogous to that employed in the quintic case.