Separatrix splitting for systems with three time scales

An exact expression for the determinant of the splitting matrix is derived for three degrees of freedom systems with three time scales: it allows us to analyze the asymptotic behaviour needed to amend the large angles theorem proposed in Ann. Inst, H. Poincare, B-60, 1 (1994). The asymptotic validity of Mel'nikov's integrals is proved for the class of models considered, which are polynomial perturbations. The technique for exhibiting cancellations is inspired by renormalization theory in quantum electrodynamics and uses an analogue of Dyson's equations to prove an infinite family of identities, due to symmetries, that remind us of Ward's identities.