Principal Poincaré–Pontryagin function associated to polynomial perturbations of a product of (d+1) straight lines

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 , with 1⩽i