The problem of center for resonant singular points of polynomial vector fields

We generalize the notion of “center” to the case of a p : −q resonant singular point of a polynomial vector field in 2 and to some other situations (resonant node, saddle-node, non-elementary singular point of vector field and resonant fixed point of one-dimensional complex diffeomorphism). We show some similarities and differences with the classical case. In particular, the analogue of Bautinʹs theorem does not hold. Four small amplitude limit cycles can bifurcate from the center after quadratic perturbation of a quadratic vector field with resonant center.