On the existence and convergence of formal power series solutions of nonlinear Mahler equations

As known, any formal power series solution $\phi \in C\left[\right[x\left]\right]$ of an algebraic equation is convergent, as well as that of an analytic one. We study the convergence of formal power series solutions of Mahler functional equations $F(x,y(x),y(x^{\ell }),\dots ,y(x^{{\ell }^n}\left)\right)=0$, where $\ell ⩾2$ is an integer and F is a holomorphic function near $0\in C^{n+2}$. Extending Bézivin's theorem from the polynomial case to the case under consideration we prove that all such solutions are also convergent. The Newton polygonal method for finding them is explained.