Bounding the length of iterated integrals of the first nonzero melnikov function

We consider small polynomial deformations of integrable systems of the form dF = 0, F is an element of C[x, y] and the first nonzero term M-mu of the displacement function Delta(t, epsilon) = Sigma(i=mu) M-i(t)is an element of(i) along a cycle gamma(t) is an element of F-1 (t). It is known that M-mu is an iterated integral of length at most mu. The bound mu depends on the deformation of dF.

In this paper we give a universal bound for the length of the iterated integral expressing the first nonzero term M-mu depending only on the geometry of the unperturbed system dF = 0. The result generalizes the result of Gavrilov and They providing a sufficient condition for M-mu to be given by an abelian integral, i.e., by an iterated integral of length 1. We conjecture that our bound is optimal.