The Center Variety of Polynomial Differential Systems

We investigate the symmetry component of the center variety of polynomial differential systems, corresponding to systems with an axis of symmetry in the real plane. We give a general algorithm to find this irreducible subvariety and compute its dimension. We show that our methods provide a simple way to compute the radical of the ideal generated by the focus quantities and, therefore, to estimate the cyclicity of a center in the case when the ideal is radical. In particular, we use our methods to get a simple proof of the famous Bautin theorem on the cyclicity of the quadratic system.