Solving the sextic by iteration: A complex dynamical approach

Recently, P. Doyle and C. McMullen devised an iterative solution to the fifth degree polynomial [4]. At the method’s core is a rational mapping f of CP with the icosahedral (A5) symmetry of a general quintic. Moreover, this A5-equivariant posseses nice dynamics: for almost any initial point a0 ∈ CP, the sequence of iterates f(a0) converges to one of the periodic cycles that comprise an icosahedral orbit. . This breaking of A5symmetry 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 points from which one computes a root. An algorithm that solves the sixth-degree equation requires a dynamical system with A6 symmetry. Since there is no action of A6 on CP, attention turns to higher dimensions. Acting onCP is anA6-isomorphic group of projective transformations that was found by Valentiner [9] in the late nineteenth century. The present work exploits this 2-dimensional A6 soccer ball in finding 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.