A Galoisian proof of Ritt theorem on the differential transcendence of Poincar\'e functions

TL;DR

This paper offers a new Galois-theoretic proof of Ritt's theorem on the differential transcendence of Poincaré functions, extending understanding of solutions to specific functional equations.

Contribution

It provides a novel Galoisian approach to prove Ritt's theorem and extends results to certain algebraic functions R.

Findings

New Galoisian proof of Ritt's theorem

Extension to algebraic functions R

Clarification of differential transcendence conditions

Abstract

Using Galois theory of functional equations, we give a new proof of the main result of the paper "Transcendental transcendency of certain functions of Poincar\'e" by J.F. Ritt, on the differential transcendence of the solutions of the functional equation R(y(t))=y(qt), where R is a rational function with complex coefficients which verifies R(0)=0, R'(0)=q, where q is a complex number with |q|>1. We also give a partial result in the case of an algebraic function R.