Abstract

In this paper, we study small polynomial perturbations of a Hamiltonian vector field with Hamiltonian F formed by a product of (d + 1) real linear functions in two variables. We assume that the corresponding lines are in a general position in R2. That is, the lines are distinct, non-parallel, no three of them have a common point and all critical values not corresponding to intersections of lines are distinct. We prove in this paper that the principal Poincaré-Pontryagin function Mk (t), associated to such a perturbation and to any family of ovals surrounding a singular point of center type, belongs to the C [t, 1 / t]-module generated by Abelian integrals and some integrals Ii, j * (t), with 1 = i < j = d defined in the paper. Moreover, Ii, j * (t) are not Abelian integrals. They are iterated integrals of length two. © 2008 Elsevier Inc. All rights reserved.