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

TL;DR

This paper investigates the convergence properties of normal forms for vector fields with nilpotent linear parts, demonstrating Gevrey regularity and the existence of an optimal truncation order that yields exponentially small remainders.

Contribution

It establishes Gevrey-$1+eta$ normal forms for vector fields with nilpotent linear parts and proves the existence of an optimal truncation order for exponentially small remainders.

Findings

Normal forms can be achieved with Gevrey-$1+eta$ regularity.

An optimal truncation order exists for the normal form procedure.

Remainders become exponentially small when truncating at this optimal order.

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.