JP Journal of Algebra, Number Theory and Applications
Volume 33, Issue 2, Pages 113 - 132
(June 2014)
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RATIONAL CYCLES OF QUADRATIC POLYNOMIALS
Samuli Piipponen and Timo Erkama
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Abstract: A new combinatorial proof of the fact that the field does not contain 4-cycles of quadratic polynomials is presented. We show that the primes dividing the common denominator of points of such a cycle appear in different configurations and that the cycle could be parametrized by seven relatively prime Gaussian integers satisfying a system of twelve algebraic equations. This system can then be analyzed by methods of computational algebraic geometry. The same idea leads to a new parametrization of rational 3-cycles and an associated reformulation of the abc-conjecture. Moreover, the method of the proof generalizes for general n-cycles as well, and as such the method provides a platform for the proof for non existence for nontrivial rational n-cycles. |
Keywords and phrases: polynomial iteration, combinatorics, algebraic geometry, rational cycle, configuration matrix, abc-conjecture. |
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