Principal Poincaré–Pontryagin function associated to polynomial perturbations of a product of [formula omitted] straight lines

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 R 2 . 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 M k ( 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 I i , j ∗ ( t ) , with 1 ⩽ i < j ⩽ d defined in the paper. Moreover, I i , j ∗ ( t ) are not Abelian integrals. They are iterated integrals of length two.