Higher order Poincare-Pontryagin functions and iterated path integrals

Abstract

We prove that the higher order Poincare-Pontryagin functions associated to the perturbed polynomial foliation $df-\epsilon (Pdx+Qdy)=0$ satisfy a differential equation of Fuchs type.

This is a revised version containing the following changes: reference [18] is added, some notations are changed, the proof of Theorem 1 is corrected, Theorem B1 and B2 are replaced now by Theorem B, some signs are corrected in the Appendix. The paper will appear in "Annales de la Faculte des Sciences de Toulouse", 2005