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

Abstract

Let L=∑_{i=1}^d X_i(z) ∂_{z_i} be a holomorphic vector field degenerating at z=0 such that Jacobi matrix ((∂ X_i/∂ z_j)(0)) has zero eigenvalues. Consider Lu=F(z, u) and let ˜{u}(z) be a formal power series solution. We study the Borel summability of ˜{u}(z), which implies the existence of a genuine solution u(z) such that u(z)∼ ˜{u}(z) as z → 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.