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

Scott Crass

doi:10.3934/jmd.2011.5.397

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2011, Volume 5, Issue 2: 397-408. Doi: 10.3934/jmd.2011.5.397

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

Scott Crass^1,

1.
Mathematics Department, The California State University at Long Beach, Long Beach, CA 90840-1001

Received: October 31, 2010

Revised: March 31, 2011

Published: June 2011

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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.

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Mathematics Subject Classification: 37F10.

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References

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