Rational functions with linear relations

TL;DR

This paper characterizes all polynomial and rational functions over a field K that satisfy a specific functional equation involving linear functions, providing a complete classification in the algebraically closed case.

Contribution

It offers a complete solution to the functional equation for polynomials and rational functions with linear relations over algebraically closed fields.

Findings

Classified all polynomial solutions to g and h linear with f(g(x))=h(f(x))

Extended the classification to rational functions over algebraically closed fields

Provided explicit forms of functions satisfying the functional equation

Abstract

We find all polynomials f,g,h over a field K such that g and h are linear and f(g(x))=h(f(x)). We also solve the same problem for rational functions f,g,h, in case the field K is algebraically closed.