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

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April 2011, 5(2): 397-408. doi: 10.3934/jmd.2011.5.397

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 November 2010 Revised April 2011 Published July 2011

Abstract
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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.

Keywords: Valentiner group, equivariant map, critically finite., complex dynamics, complex reflection group.

Mathematics Subject Classification: 37F1.

Citation: Scott Crass. New light on solving the sextic by iteration: An algorithm using reliable dynamics. Journal of Modern Dynamics, 2011, 5 (2) : 397-408. doi: 10.3934/jmd.2011.5.397

References:

[1]	S. Crass and P. Doyle, Solving the sextic by iteration: A complex dynamical approach, Internat. Math. Res. Notices, (1997), 83-99. doi: 10.1155/S1073792897000068.
[2]	S. Crass, Solving the sextic by iteration: A study in complex geometry and dynamics, Experiment. Math., 8 (1999), 209-240. Preprint at: http://arxiv.org/abs/math.DS/9903111.
[3]	S. Crass, A family of critically finite maps with symmetry, Publ. Mat., 49 (2005), 127-157. Preprint at: http://arxiv.org/abs/math.DS/0307057.
[4]	S. Crass, Solving the heptic by iteration in two dimensions: Geometry and dynamics under Klein's group of order 168, J. Mod. Dyn., 1 (2007), 175-203. doi: 10.3934/jmd.2007.1.175.
[5]	, S. Crass, Available from: http://www.csulb.edu/~scrass/math.html
[6]	J. E. Fornaess and N. Sibony, Complex dynamics in higher dimension I, Complex analytic methods in dynamical systems (Rio de Janeiro, 1992), Astérisque, 222 (1994), 201-231.
[7]	C. McMullen, Julia, (computer program). Available from: http://www.math.harvard.edu/~ctm/programs.html
[8]	H. Nusse and J. Yorke, "Dynamics: Numerical Explorations," Second edition. Accompanying computer program Dynamics 2 coauthored by Brian R. Hunt and Eric J. Kostelich, With 1 IBM-PC floppy disk (3.5 inch; HD). Applied Mathematical Sciences, 101. Springer-Verlag, New York, 1998.
[9]	G. Shephard and T. Todd, Finite unitary reflection groups, Canad. J. Math., 6 (1954), 274-304. doi: 10.4153/CJM-1954-028-3.
[10]	K. Ueno, Dynamics of symmetric holomorphic maps on projective spaces, Publ. Mat., 51 (2007), 333-344.

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References:

[1]	S. Crass and P. Doyle, Solving the sextic by iteration: A complex dynamical approach, Internat. Math. Res. Notices, (1997), 83-99. doi: 10.1155/S1073792897000068.
[2]	S. Crass, Solving the sextic by iteration: A study in complex geometry and dynamics, Experiment. Math., 8 (1999), 209-240. Preprint at: http://arxiv.org/abs/math.DS/9903111.
[3]	S. Crass, A family of critically finite maps with symmetry, Publ. Mat., 49 (2005), 127-157. Preprint at: http://arxiv.org/abs/math.DS/0307057.
[4]	S. Crass, Solving the heptic by iteration in two dimensions: Geometry and dynamics under Klein's group of order 168, J. Mod. Dyn., 1 (2007), 175-203. doi: 10.3934/jmd.2007.1.175.
[5]	, S. Crass, Available from: http://www.csulb.edu/~scrass/math.html
[6]	J. E. Fornaess and N. Sibony, Complex dynamics in higher dimension I, Complex analytic methods in dynamical systems (Rio de Janeiro, 1992), Astérisque, 222 (1994), 201-231.
[7]	C. McMullen, Julia, (computer program). Available from: http://www.math.harvard.edu/~ctm/programs.html
[8]	H. Nusse and J. Yorke, "Dynamics: Numerical Explorations," Second edition. Accompanying computer program Dynamics 2 coauthored by Brian R. Hunt and Eric J. Kostelich, With 1 IBM-PC floppy disk (3.5 inch; HD). Applied Mathematical Sciences, 101. Springer-Verlag, New York, 1998.
[9]	G. Shephard and T. Todd, Finite unitary reflection groups, Canad. J. Math., 6 (1954), 274-304. doi: 10.4153/CJM-1954-028-3.
[10]	K. Ueno, Dynamics of symmetric holomorphic maps on projective spaces, Publ. Mat., 51 (2007), 333-344.

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