Journal of Modern Dynamics (JMD)

Solving the heptic by iteration in two dimensions: Geometry and dynamics under Klein's group of order 168

Pages: 175 - 203, Issue 2, April 2007      doi:10.3934/jmd.2007.1.175

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Scott Crass - Mathematics Department, The California State University at Long Beach, Long Beach, CA 90840-1001, United States (email)

Abstract: There is a family of seventh-degree polynomials $H$ whose members possess the symmetries of a simple group of order $168$. This group has an elegant action on the complex projective plane. Developing some of the action's rich algebraic and geometric properties rewards us with a special map that also realizes the $168$-fold symmetry. The map's dynamics provides the main tool in an algorithm that solves certain "heptic" equations in $H$.

Keywords:  complex dynamics,equivariant map, reflection group,seventh-degree equation.
Mathematics Subject Classification:  37C30, 37D30.

Received: September 2006;      Available Online: January 2007.