We study foliations in $\mathbb{C}^2$ given by polynomial deformations of the form $dH+\epsilon \eta=0$, with $\gamma(t)\subset H^{-1}(t)$ a family of cycles. The Poincaré first return map is of the form $P(t)=t+\sum_j \epsilon^j M_j^\gamma(t).$ The functions $M_j^\gamma$ are called Melnikov functions and are given by iterated integrals of orbit length at most $j$. This length is a measure of the complexity of Melnikov functions.

We show that, for each $k\in\mathbb{N}$, there exists a universal Noetherianity index $n_{ H,\gamma}(k)$, independent of the deformation $\eta$, such that, if $M_j^\gamma=0$, for $j=1,\ldots,n_{ H,\gamma}(k)$, then $M_j^\gamma$ is of orbit length $j-k$, for any Melnikov function $M_j^\gamma$.

In order to prove this theorem, we develop a structure theorem for Melnikov functions and use the Ritt-Raudenbush differential algebra theorem.

We calculate the universal Noetherianity index $n_{H,\gamma}(k)$ in various nontrivial examples.

The presented work is a recent work which is a continuation of the work to be presented here by J. Pontigo-Herrera.