Building counterexamples to generalizations for rational functions of Ritt's decomposition theorem

TL;DR

This paper constructs explicit counterexamples to Ritt's first theorem for rational functions, showing that decomposition chain lengths can differ, and introduces an algorithm for computing the fixing group of such functions.

Contribution

It provides the first known counterexamples to Ritt's theorem for rational functions and develops an algorithm for computing the fixing group over Q.

Findings

Counterexamples with different decomposition chain lengths for rational functions.

An algorithm for computing the fixing group of a rational function over Q.

Insights into the stability of the base field in rational function decomposition.

Abstract

The classical Ritt's Theorems state several properties of univariate polynomial decomposition. In this paper we present new counterexamples to Ritt's first 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 Q 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 A_4 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 Q.