Invariance of Poincare-Lyapunov polynomials under the group of rotations

We show that the Poincar'e-Lyapunov polynomials at a focus of a family of real polynomial vector fields of degree $n$ on the plane are invariant under the group of rotations. Furthermore, we show that under the multiplicative group ${Bbb C}^*={ ho { m e}^{ipsi}}$, they are invariant up to a positive factor. These results follow from the weighted-homogeneity of the polynomials that we define in the text.