New light on solving the sextic by iteration: An algorithm using reliable dynamics

Abstract

In recent work on holomorphic maps that are symmetric under certain complex reflection groups---generated by complex reflections through a set of hyperplanes, the author announced a general conjecture related to reflection groups. The claim is that for each reflection group G, there is a G-equivariant holomorphic map that is critical exactly on the set of reflecting hyperplanes. One such group is the Valentiner action V---isomorphic to the alternating group A_6---on the complex projective plane. A previous algorithm that solved sixth-degree equations harnessed the dynamics of a V-equivariant. However, important global dynamical properties of this map were unproven. Revisiting the question in light of the reflection group conjecture led to the discovery of a degree-31 map that is critical on the 45 lines of reflection for V. The map's critical finiteness provides a means of proving its possession of the previous elusive global properties. Finally, a sextic-solving procedure that employs this map's reliable dynamics is developed.