Hilbert's 16th problem for quadratic systems and cyclicity of elementary graphics

Abstract

In this paper we study the finite cyclicity of several elementary graphics appearing in quadratic systems. This makes substantial progress in the study of the finite cyclicity of the elementary graphics with non-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 normal forms for elementary singular points in the graphics. Several arguments use the fact that two singular points `compensate' each other precisely when the graphic surrounds a centre. One originality of the paper is to prove that for certain graphics among quadratic systems some regular transition maps are not tangent to the identity.