Normal forms with exponentially small remainder and Gevrey normalization for vector fields with a nilpotent linear part

Abstract

We explore the convergence/divergence of the normal form for a singularity of a vector field on $\C^n$ with nilpotent linear part. We show that a Gevrey-$\alpha$ vector field $X$ with a nilpotent linear part can be reduced to a normal form of Gevrey-$1+\alpha$ type with the use of a Gevrey-$1+\alpha$ transformation. We also give a proof of the existence of an optimal order to stop the normal form procedure. If one stops the normal form procedure at this order, the remainder becomes exponentially small.