Solving the sextic by iteration: A complex dynamical approach
Abstract
Recently, Peter Doyle and Curt McMullen devised an iterative solution to the fifth degree polynomial. At the method's core is a rational mapping of the Riemann sphere with the icosahedral symmetry of a general quintic. Moreover, this map posseses "reliable" dynamics: for almost any initial point, the its trajectory converges to one of the periodic cycles that comprise an icosahedral orbit. This symmetrybreaking provides for a reliable or "generallyconvergent" quinticsolving algorithm: with almost any fifthdegree equation, associate a rational mapping that has reliable dynamics and whose attractor consists of points from which one computes a root. An algorithm that solves the sixthdegree equation requires a dynamical system with the symmetry of the alternating group on six things. This group does not act on the Riemmann sphere, but does act on the complex projective planethis is the Valentiner group. The present work exploits the resulting 2dimensional geometry in finding a Valentinersymmetric rational mapping whose elegant dynamics experimentally appear to be reliable in the above sensetransferred to the 2dimensional setting. This map provides the central feature of a conjecturallyreliable sexticsolving algorithm analogous to that employed in the quintic case.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 March 1999
 DOI:
 10.48550/arXiv.math/9903106
 arXiv:
 arXiv:math/9903106
 Bibcode:
 1999math......3106C
 Keywords:

 Dynamical Systems
 EPrint:
 19 pages, 2 figures