Artículo de publicación ISIIn 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.