Solving the quintic by iteration in three dimensions

Details Solving the quintic by iteration in three dimensions Solving the quintic by iteration in three dimensions

1999-03-09

arxiv.org/abs/math/9903054v2

Mathematics

Dynamical Systems

40 pages, 15 figures

Scientific paper

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 S_5. Induced by its five-dimensional linear permutation representation is a three-dimensional projective action. A mapping of complex projective 3-space with this S_5 symmetry can provide the requisite symmetry-breaking tool. The article describes some of the S_5 geometry in CP^3 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 - three 1-dimensional iterations, the present solution employs one 3-dimensional iteration.

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