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.

polynomial iteration, combinatorics, algebraic geometry, rational cycle, configuration matrix, abc-conjecture.