Abstract
The center problem and the cyclicity of singular points of the Kukles system are studied. The necessary conditions for the center at the origin are obtained as the variety of the ideal of Lyapunov quantities calculated by direct solution of the polynomial system whose left-hand sides form the Gröbner basis of the ideal. This ideal is also used to calculate the cyclicity of the centers and foci of the system. A theorem is proved that allows one to find the cyclicity of the centers of polynomial systems by using its Gröbner basis instead of the ideal of Lyapunov quantities.
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