The classical Ritt's Theorems state several properties of univariatepolynomial decomposition. In this paper we present new counterexamples toRitt's first theorem, which states the equality of length of decompositionchains of a polynomial, in the case of rational functions. Namely, we providean explicit example of a rational function with coefficients in Q and twodecompositions of different length. Another aspect is the use of some techniques that could allow for othercounterexamples, namely, relating groups and decompositions and using the factthat the alternating group A_4 has two subgroup chains of different lengths;and we provide more information about the generalizations of another propertyof polynomial decomposition: the stability of the base field. We also presentan algorithm for computing the fixing group of a rational function providingthe complexity over Q.
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