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 L u = 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.