Formal Gevrey solutions -- in analytic germs -- for higher order holomorphic PDEs

Abstract

We establish existence, uniqueness and the Gevrey order of formal solutions of a wide family of systems of holomorphic PDEs. These equations exhibit a singular locus given by the zero set of an analytic map P with P (0) = 0, which is the source of divergence of the solution. Examples where the Gevrey bound is optimal are provided. Moreover, we give Poincaré type conditions to recover convergence solutions when P is not singular at 0.