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

We consider small polynomial deformations of integrable systems of the form $dF=0$, $F\in \mathbb{C}[x,y]$ and the first nonzero term $M_{\mu }$ of the displacement function $\mathrm{\Delta }(t,ϵ)={\sum }_{i=\mu }{M}_i(t){ϵ}^i$ 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 $\mu$ 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 topology 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.