Rational functions admitting double decomposition

TL;DR

This paper explores a class of rational functions, including Zolotarev fractions, that admit double decompositions, extending Ritt's polynomial decomposition theory to new rational function classes with symmetric representations.

Contribution

It introduces a new class of rational functions with double decompositions, providing a symmetric representation similar to Chebyshev polynomials, expanding the understanding of rational function decomposition.

Findings

Identified a class of rational functions with double decompositions.

Provided a symmetric parametric representation of Zolotarev fractions.

Connected Zolotarev fractions to Chebyshev polynomials as a special case.

Abstract

J.Ritt has investigated the structure of complex polynomials with respect to superposition. In particular, he listed all the polynomials admitting different double decompositions into indecomposable polynomials. The analogues of Ritt theory for rational functions were constructed just for several particular classes of the said functions, say for Laurent polynomials (F.Pakovich). In this note we describe a certain class of double decompositions for rational functions. Essentially, described below rational functions were discovered by E.I.Zolotarev in 1877 as a solution of certain optimization problem. However, the double decomposition property for them was hidden until recently because of somewhat awkward representation. We give a (possibly new) symmetric representation of Zolotarev fractions resembling the parametric representation for Chebyshev polynomials, which are a special limit case of Zolotarev fraction.