A Galoisian proof of Ritt theorem on the differential transcendence of Poincaré functions

with Gwladys Fernandes 

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 Poincaré'' by J.F.~Ritt, on the differential transcendence of the solutions of the functional equation $R(y(t))=y(qt)$, where $R(t)\in\mathbb C(t)$ verifies $R(0)=0$, $R'(0)=q\in\mathbb  C$, with $|q|>1$.
We also give a partial result in the case of an algebraic function $R$.

DOI: 

ArXiv: 2102.08268

HAL: hal-03146286