Principal part of multi-parameter displacement functions

In this paper we investigate planar polynomial multi-parameter deformations of Hamiltonian vector fields. We study first all coefficients in the development of the displacement function on a transversal to the period annulus. We show that they can be expressed through iterated integrals, whose length is bounded by the degree of the monomials.A second result expresses the principal terms in the division of the displacement function in the Bautin ideal. More precisely, the principal terms in its division in a reduced basis of the Bautin ideal are given by iterated integrals. Our approach is algorithmic and generalizes Françoise algorithm for one-parameter families.