2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/59877We 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.A new version, extended introduction and more referencesDynamical SystemsRings and Algebrastext