Normal forms for vector fields with respect to an arbitrary dilation

Homogenous vector fields of degree one with respect to an arbitrary dilation ^r may be regarded as a natural extension of the linear vector fields. This paper is concerned with the problem of computing normal forms for vector fields that can be expanded in terms of homogeneous fields of degree greater than or equal to one with respect to an arbitrary dilation ^r. In particular, necessary and sufficient conditions are given for the existence of a local coordinate change that transforms an analytic field into a homogeneous field of degree one with respect to an arbitrary dilation ^r.This research was partially supported by National Science Foundation (Grant DMS-9301039), and by Istituto Nazionale di Alta Matematica F. Severi.