Borel summability of formal solutions of some first order singular partial differential equations and normal forms of vector fields

Abstract

Let $L={\sum }_{i=1}^d{X}_i\left(z\right){\partial }_{z_i}$ be a holomorphic vector field degenerating at $z=0$ such that Jacobi matrix$\left(\right(\partial X_i/\partial z_j\left)\right(0\left)\right)$ has zero eigenvalues. Consider $Lu=F(z,u)$ and let $\tilde{u}\left(z\right)$ be a formal power series solution. We study the Borel summability of $\tilde{u}\left(z\right)$, which implies the existence of a genuine solution $u\left(z\right)$ such that $u\left(z\right)\sim \tilde{u}\left(z\right)$ as $z\to 0$ in some sectorial region. Further we treat singular equations appearing in finding normal forms of singular vector fields and study to simplify $L$ by transformations with Borel summable functions.