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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 908401001 
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 sixthdegree 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 degree31 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 sexticsolving procedure that employs this map's reliable dynamics is developed.
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/CJM19540283. 
[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/CJM19540283. 
[10] 
K. Ueno, Dynamics of symmetric holomorphic maps on projective spaces,, Publ. Mat., 51 (2007), 333. 
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