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