This paper is a part of the proof of the existential part of Hilbert's 16th problem for quadratic vector fields initiated in . Its principal aim is the proof of the finite cyclicity of four elementary graphics among quadratic systems with two semi-hyperbolic points and surrounding a center, namely, the graphics (I52, (I91), (H52) and (H111) (using the names introduced in ). The technique used is a refinement of the technique of Khovanskii as adapted to the finite cyclicity of graphics by Il'yashenko and Yakovenko, together with an equivalent of the Bautin trick to treat the center case. We show that the cyclicity of each of the first three graphics is equal to 2 and that the cyclicity of the fourth one is equal to three. We improve the known results about finite cyclicity of the graphics (H81), (H101), (I42) by showing that their cyclicities are equal to 2.
Polynomial vector fieldsfinite number of limit cyclesfinite cyclicity of graphicssemi-hyperbolic pointsKhovanskii procedure