On the existence and convergence of formal power series solutions of nonlinear Mahler equations (Q6650586)

The linear Mahler equation has the following form:\N\[\Na_0(x)y(x)+a_1(x)y(x^{\ell})+\cdots+a_n(x)y(x^{\ell n})=b(x), \qquad \ell\in \mathbb{N}, \quad \ell\ge 2,\N\]\Nwhere \(a_0,\cdots,a_n\) and \(b\) are polynomial in \(\mathbb{C}[x]\).\N\NA formal power series \(\phi(x)\in \mathbb{C}[[x]]\) is said to be an \(\ell\)-Mahler function if it satisfies the previous equation.\N\NIn this paper, the authors consider the nonlinear Mahler functional equation\N\[\NF(x,y,\mu y,\cdots,\mu^ny)=0, \tag{1}\N\]\Nwhere \(\mu y(x)=y(x^{\ell})\) and \(F(x,y_0,y_1,\cdots,y_n)\) is a nonzero holomorphic function near \(0\in \mathbb{C}^{n+2}\), and investigate the question of existence and convergence of formal power series solutions of the equation.\N\NThe following theorem has been proved in [\textit{J.-P. Bézivin}, Funkc. Ekvacioj, Ser. Int. 37, No. 2, 263--271 (1994; Zbl 0810.39006)]: \N\NTheorem. Any formal power series solution of the equation \(P(x,y,\mu y,\cdots,\mu^ny)=0\), where \(P(x,y_0,y_1,\cdots,y_n)\) is a nonzero polynomial with holomorphic coefficients near \(x=0\), has a nonzero radius of convergence.\N\NThe main result of the present paper is given by the following claim. \N\NTheorem. If the formal power series \(\phi(x)=\sum_{k=0}^{\infty} c_kx^k \in \mathbb{C}[[x]]\) satisfies (1) then it has a nonzero radius of convergence.