Finite cyclicity of nilpotent graphics of pp-type surrounding a center

Abstract

This paper is part of the DRR-program of [4] to prove the finiteness part of Hilbert's 16th problem for quadratic vector fields by showing the finite cyclicity of 121 graphics. In this paper we prove the finite cyclicity of 4 graphics passing through a triple nilpotent point of elliptic type surrounding a center, namely the graphics $(H_7^1)$, $(F_{7a}^1)$, $(H_{11}^3)$ and $(I_{6a}^1)$. These four graphics are of pp-type, in the sense that they join two parabolic sectors of the nilpotent point. The exact cyclicity is 2 for $(H_7^1)$ and $(H_{11}^3)$. The graphics $(F_{7a}^1)$ and $(I_{6a}^1)$ occur in continuous families. Their exact cyclicity is 2 except for a discrete subset of such graphics. The method can be applied to most other graphics of the DRR-program [4] through a triple nilpotent point and surrounding a center.