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

Abstract

There is a family of seventh-degree polynomials $H$ whose memberspossess the symmetries of a simple group of order $168$. This grouphas an elegant action on the complex projective plane. Developingsome of the action's rich algebraic and geometric properties rewardsus with a special map that also realizes the $168$-fold symmetry.The map's dynamics provides the main tool in an algorithm thatsolves certain 'heptic' equations in $H$.