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

1. 

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

Received  November 2010 Revised  April 2011 Published  July 2011

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.
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. doi: 10.1155/S1073792897000068.

[2]

S. Crass, Solving the sextic by iteration: A study in complex geometry and dynamics,, Experiment. Math., 8 (1999), 209.

[3]

S. Crass, A family of critically finite maps with symmetry,, Publ. Mat., 49 (2005), 127.

[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. doi: 10.3934/jmd.2007.1.175.

[5]

, S. Crass,, Available from: \url{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, 222 (1994), 201.

[7]

C. McMullen, Julia, (computer program)., Available from: \url{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, (1998).

[9]

G. Shephard and T. Todd, Finite unitary reflection groups,, Canad. J. Math., 6 (1954), 274. doi: 10.4153/CJM-1954-028-3.

[10]

K. Ueno, Dynamics of symmetric holomorphic maps on projective spaces,, Publ. Mat., 51 (2007), 333.

show all references

References:
[1]

S. Crass and P. Doyle, Solving the sextic by iteration: A complex dynamical approach,, Internat. Math. Res. Notices, (1997), 83. doi: 10.1155/S1073792897000068.

[2]

S. Crass, Solving the sextic by iteration: A study in complex geometry and dynamics,, Experiment. Math., 8 (1999), 209.

[3]

S. Crass, A family of critically finite maps with symmetry,, Publ. Mat., 49 (2005), 127.

[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. doi: 10.3934/jmd.2007.1.175.

[5]

, S. Crass,, Available from: \url{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, 222 (1994), 201.

[7]

C. McMullen, Julia, (computer program)., Available from: \url{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, (1998).

[9]

G. Shephard and T. Todd, Finite unitary reflection groups,, Canad. J. Math., 6 (1954), 274. doi: 10.4153/CJM-1954-028-3.

[10]

K. Ueno, Dynamics of symmetric holomorphic maps on projective spaces,, Publ. Mat., 51 (2007), 333.

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