Journal of Modern Dynamics (JMD)

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

Pages: 397 - 408, Issue 2, April 2011      doi:10.3934/jmd.2011.5.397

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Scott Crass - Mathematics Department, The California State University at Long Beach, Long Beach, CA 90840-1001, United States (email)

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 $\mathcal{V}$---isomorphic to the alternating group $\mathcal{A}_6$---on the complex projective plane. A previous algorithm that solved sixth-degree equations harnessed the dynamics of a $\mathcal{V}$-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 $\mathcal{V}$. 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.

Keywords:  Valentiner group, complex reflection group, equivariant map, complex dynamics, critically finite.
Mathematics Subject Classification:  37F10.

Received: November 2010;      Revised: April 2011;      Available Online: July 2011.