Building counterexamples to generalizations for rational functions of Rittʹs decomposition theorem

The classical Rittʹs theorems state several properties of univariate polynomial decomposition. In this paper we present new counterexamples to the First Ritt Theorem, which states the equality of length of decomposition chains of a polynomial, in the case of rational functions. Namely, we provide an explicit example of a rational function with coefficients in and two decompositions of different length. Another aspect is the use of some techniques that could allow for other counterexamples, namely, relating groups and decompositions and using the fact that the alternating group A4 has two subgroup chains of different lengths; and we provide more information about the generalizations of another property of polynomial decomposition: the stability of the base field. We also present an algorithm for computing the fixing group of a rational function providing the complexity over the rational number field.