On the equation P(f)=Q(g), where P,Q are polynomials and f,g are entire functions

TL;DR

This paper extends the classification of solutions to the functional equation P(f)=Q(g) by allowing entire functions f,g and rational functions P,Q, also addressing strong uniqueness polynomials for entire functions.

Contribution

It generalizes Ritt's classical results from polynomials to entire functions and rational functions, providing a comprehensive description of solutions and strong uniqueness polynomials.

Findings

Classified solutions for s=P(f)=Q(g) with entire functions and rational functions.

Identified conditions for strong uniqueness polynomials for entire functions.

Extended polynomial solution theory to broader classes of functions.

Abstract

In 1922 Ritt described polynomial solutions of the functional equation P(f)=Q(g). In this paper we describe solutions of the equation above in the case when P,Q are polynomials while f,g are allowed to be arbitrary entire functions. In fact, we describe solutions of the more general functional equation s=P(f)=Q(g), where s,f,g are entire functions and P,Q are arbitrary rational functions. Besides, we solve the problem of description of "strong uniqueness polynomials" for entire functions.