Solving the octic by iteration in six dimensions

Abstract

The requirement for solving a polynomial is a means of breaking its symmetry, which in the case of the octic is that of the symmetric group S 8 . Its eight-dimensional linear permutation representation restricts to a six-dimensional projective action. A mapping of complex projective 6-space with this S 8 symmetry can provide the requisite symmetry-breaking tool. This paper describes some of the S 8 geometry in CP 6 as well as a special S 8 -symmetric rational map in degree four. Several basins-of-attraction plots illustrate the map's geometric and dynamical properties. The work culminates with an explicit algorithm that uses this map to solve a general octic. A concluding discussion treats the generality of this approach to equations in higher degree.