Study of the cyclicity of some degenerate graphics inside quadratic systems

Freddy Dumortier; Christiane Rousseau

doi:10.3934/cpaa.2009.8.1133

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2009, Volume 8, Issue 4: 1133-1157. Doi: 10.3934/cpaa.2009.8.1133

This issue Previous Article Next Article On a class of hypoelliptic operators with unbounded coefficients in $R^N$

Study of the cyclicity of some degenerate graphics inside quadratic systems

Freddy Dumortier^1, and
Christiane Rousseau^2,

1.
Universiteit Hasselt, Campus Diepenbeek, Agoralaan–gebouw D, 3590 Diepenbeek
2.
DMS and CRM, Université de Montréal

Received: December 31, 2007

Revised: October 31, 2008

Published: June 2009

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Abstract

In this paper we make essential steps in proving the finite cyclicity of degenerate graphics in quadratic systems, having a line of singular points in the finite plane. In particular we consider the graphics $(DF_{1 a})$, $(DF_{2 a})$ of the program of [8] to prove the finiteness part of Hilbert's 16th problem for quadratic vector fields. We make a complete treatment except for one very specific problem that we clearly identify.

Keywords:

Finite cyclicity,
singular perturbations,
blow-up of the family,
degenerate graphics,
Hilbert’s 16-th problem for quadratic systems.

Mathematics Subject Classification: 34C07, 34C26, 37G99.

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