Normal forms for vector fields with respect to an arbitrary dilation (Q1922833)

The problem of computing normal forms for vector fields having a (possibly degenerate) linear part is considered. The main idea is to consider the given vector field as a perturbation of its homogeneous part of degree one with respect to some dilation \(\delta^r_\varepsilon\). The main result of the paper gives a necessary and sufficient condition for the existence of a local, analytic coordinate change which transforms a real analytic vector field into a homogeneous field of degree greater than or equal to one with respect to a given dilation \(\delta^r_\varepsilon\). An example illustrating how this result can be used to compute a coordinate change which transforms a given field into a homogeneous field of degree one with respect to a dilation \(\delta^r_\varepsilon\) is given.