The infinitesimal 16th Hilbert problem in the quadratic case

Abstract.

Let H(x,y) be a real cubic polynomial with four distinct critical values (in a complex domain) and let ${X_H} = {H_y}\frac{\partial }{{\partial x}} - {H_x}\frac{\partial }{{\partial y}}$ be the corresponding Hamiltonian vector field. We show that there is a neighborhood ? of X _H in the space of all quadratic plane vector fields, such that any X∈? has at most two limit cycles.