Rational functions admitting double decompositions

Abstract:

Ritt (1922) studied the structure of the set of complex polynomials with respect to composition. A polynomial $P(x)$ is said to be indecomposable if it can be represented as $P=P_1\circ P_2$ only if either $P_1$ or $P_2$ is a linear function. A decomposition $P=P_1\circ P_2\circ \ldots \circ P_r$ is said to be maximal if all the $P_j$ are indecomposable polynomials that are not linear. Ritt proved that any two maximal decompositions of the same polynomial have the same length $r$, the same (unordered) set $\{\deg (P_j)\}$ of the degrees of the composition factors, and can be connected by a finite chain of transformations each of which consists in replacing the left-hand side of the double decomposition \begin{equation} R_1\circ R_2=R_3\circ R_4 \end{equation} by its right-hand side. Solutions of this functional equation are indecomposable polynomials of degree greater than 1, and Ritt listed all of them explicitly.

Up until now, analogues of Ritt’s theory for rational functions have only been constructed for some special classes of these functions, for instance, for Laurent polynomials (Pakovich, 2009). In this note we describe a certain class of double decompositions (\ref{DD1}) with rational functions $R_j(x)$ of degree greater than 1. In essence, the rational functions described below were discovered by Zolotarëv as solutions of a certain optimization problem (1932). However, the double decomposition property for these functions remained little known because they had an awkward parametric representation. Below we give a representation for Zolotarëv fractions (possibly new), which resembles the well-known representation for Chebyshëv polynomials. These, by the way, are a special limit case of Zolotarëv fractions.