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

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  • 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.
    Mathematics Subject Classification: 37F10.


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