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Invariance of Poincare-Lyapunov Polynomials Under the Group of Rotations

We show that the Poincaré-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 C = {ρe}, they are invariant up to a positive factor. These results follow from the weighted-homogeneity of the polynomials that we define in the text.