Holomorphic normal form of nonlinear perturbations of nilpotent vector fields (Q327551)

The paper is a study of the local classification of holomorphic vector fields in a neighborhood of a fixed point in \(\mathbb{C}^n\), \(n \geq 3\), having a regular nilpotent linear part (a nilpotent matrix is said to be regular if its Jordan normal form does not contain zero blocks). The main result is a sufficient condition that ensures that the germ of such a vector field can be holomorphically conjugated to its formal normal form, that is, the normalizing formal transformation converges. The authors show that this condition is a nilpotent version of the \textit{condition~A} introduced by A.D.~Bruno.