In this talk I will present some results obtained with D. Novikov, L. Ortiz-Bobadilla and J. Pontigo-Herrera.

We study small polynomial deformations of a Hamiltonian system in the plane of the form $dF+\epsilon\omega=0$. We consider a family of regular cycles $\gamma(t)\in F^{-1}(t)$ and the displacement function $\Delta(t)$ along this family on a transversal parametrized by the values $t$ of $F$.

Then $\Delta(t)=\sum_{j=\mu}M_j(t)\epsilon^j$. The functions $M_j$ are called Melnikov functions and we assume that $M_\mu\not\equiv0$. It is the first nonzero Melnikov function. It is known that $M_\mu$ is given by an iterated integral of length at most $\mu$. In general, the minimal length depends on the perturbation $\omega$.

We want to give a bound for this minimal length, independent of the perturbation $\omega$ and depending only on the Hamiltonian $F$ and the family of cycles $\gamma$.