Electron. J. Diff. Eqns., Vol. 1998(1998), No. 23, pp. 1-8.

Invariance of Poincare-Lyapunov polynomials under the group of rotations

Pierre Joyal

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.

Submitted June 25, 1998. Published October 9, 1998.
Math Subject Classification: 58F14, 58F21, 58F35, 34C25.
Key Words: focus, invariance of Poincare-Lyapunov polynomials, weighted-homogeneity.

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Pierre Joyal
Departement d'informatique et de mathematique
Universite du Quebec a Chicoutimi
555 boul. de l'Universite, Chicoutimi, G7H 2B1, Canada
e-mail: Pierre_Joyal@uqac.uquebec.caname
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