Solving the quintic by iteration in three dimensions

Abstract

The requirement for solving a polynomial is a means of breaking its symmetry, which in the case of the quintic, is that of the symmetric group S₅. Induced by its five-dimensional linear permutation representation is a three-dimensional projective action. A mapping of complex projective 3-space with this S₅ symmetry can provide the requisite symmetry-breaking tool. The article describes some of the S₅ geometry in CP³ as well as several maps with particularly elegant geometric and dynamical properties. Using a rational map in degree six, it culminates with an explicit algorithm for solving a general quintic. In contrast to the Doyle-McMullen procedure, which involves three 1-dimensional iterations, the present solution employs one 3-dimensional iteration.