Decompositions of rational functions over real and complex numbers and a question about invariant curves

TL;DR

This paper explores how rational functions over real and complex numbers can be decomposed and investigates the properties of invariant curves on a Riemann sphere under these functions.

Contribution

It introduces new insights into the decomposition of rational functions and addresses a specific question about invariant curves on the Riemann sphere.

Findings

Decomposition methods for rational functions over real and complex fields.

Characterization of invariant curves under rational functions.

Connections between function decompositions and geometric invariants.

Abstract

We consider the connection of functional decompositions of rational functions over the real and complex numbers, and a question about curves on a Riemann sphere which are invariant under a rational function.