The diophantine nature for the convergence of formal solutions (Q1075488)

The author studies the convergence of all formal solutions \(u(x)=x^{\omega}\sum_{\eta \in {\mathbb{N}}^ d}u_{\eta}x^{\eta}/\eta !\) of equations, essentially, of the form \[ Pu\in \sum_{| \alpha | =| \beta | \leq m}x^{\alpha}a_{\alpha \beta}(x)(\partial /\partial x)^{\beta}u(x)=f\quad (x)x^{\omega} \] where \(\omega \in {\mathbb{C}}^ d\), \(x\in {\mathbb{C}}^ d\), \(d\geq 2\), \(\alpha =(\alpha_ 1,...,\alpha_ d)\in {\mathbb{N}}^ d\), \({\mathbb{N}}=\{0,1,2,...\}\) and where \(a_{\alpha \beta}(x)\) is analytic at \(x=0\). He gives a necessary and sufficient condition for the convergence of all formal solutions, which is described by use of two diophantine functions.