2011/06/16 by Scott Crass, Crass, Scott
Mathematics · #37F10 #Dynamical Systems (math.DS) #FOS: Mathematics #math.DS #msc:37F10
paper · pdf · doi:10.48550/arxiv.1106.3304
To appear in Journal of Modern Dynamics
arxiv created 2011/06/16 · arxiv updated 2011/06/17
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 A6---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.