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We consider small polynomial deformations of integrable systems of the form $dF=0, F\in\mathbb{C}[x,y]$ and the first nonzero term $M_\mu$ of the displacement function $\Delta(t,\epsilon)=\sum_{i=\mu}M_i(t)\epsilon^i$ along a cycle $\gamma(t)\in F^{-1}(t)$. It is known that $M_\mu$ is an iterated integral of length at most $\mu$. The bound $\mu$ epends 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 $\mu$ depending only on the geometry of the unperturbed system $dF=0$. The result generalizes the result of Gavrilov and Iliev 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.

Bounding the Length of Iterated Integrals of the First Nonzero Melnikov Function