HILBERTS 16TH PROBLEM FOR QUADRATIC SYSTEMS AND CYCLICITY OF ELEMENTARY GRAPHICS

In this paper we study the finite cyclicity of several elementary grap hics appearing in quadratic systems. This makes substantial progress i n the study of the finite cyclicity of the elementary graphics with no n-identical return map listed in Dumortier et al J. Diff. Eqns 110 86- 133. The main tool we use is the method of Khovanskii. We also use the fact that some graphics have unbroken connections and we calculate no rmal forms for elementary singular points in the graphics. Several arg uments use the fact that two singular points 'compensate' each other p recisely when the graphic surrounds a centre. One originality of the p aper is to prove that for certain graphics among quadratic systems som e regular transition maps are not tangent to the identity.