Solving the sextic by iteration: A complex dynamical approach

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dc.creatorCrass, Scott
dc.creatorDoyle, Peter
dc.date1999-03-17
dc.date1999-03-18
dc.date.accessioned2026-07-07T05:28:21Z
dc.date.available2026-07-07T05:28:21Z
dc.descriptionRecently, 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 symmetry-breaking provides for a reliable or "generally-convergent" quintic-solving algorithm: with almost any fifth-degree 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 sixth-degree 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 plane--this is the Valentiner group. The present work exploits the resulting 2-dimensional geometry in finding a Valentiner-symmetric rational mapping whose elegant dynamics experimentally appear to be reliable in the above sense---transferred to the 2-dimensional setting. This map provides the central feature of a conjecturally-reliable sextic-solving algorithm analogous to that employed in the quintic case.
dc.description19 pages, 2 figures
dc.identifierhttps://arxiv.org/abs/math/9903106
dc.identifierhttp://arxiv.org/abs/math/9903106
dc.identifierInternational Mathematics Research Notices, 1997, No.2, 83-99
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/78231
dc.subjectDynamical Systems
dc.titleSolving the sextic by iteration: A complex dynamical approach
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