## Invariance of Poincaré-Lyapunov polynomials under the group of rotations

### Summary

Summary: We show that the Poincare-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 ${\Bbb C}^*=\{\rho {\rm e}^{i\psi}\}$, they are invariant up to a positive factor. These results follow from the weighted-homogeneity of the polynomials that we define in the text.

### Mathematics Subject Classification

58F14, 58F21, 58F35, 34C25

### Keywords/Phrases

focus, invariance of Poincarè-Lyapunov polynomials, weighted-homogeneity