  Let $F\in\mathbb{C}[x,y]$ be a polynomial, $\gamma(z)\in \pi_1(F^{-1}(z))$ a
non-trivial cycle in a generic fiber of $F$ and let $\omega$ be a polynomial
$1$-form, thus defining a polynomial deformation $dF+\epsilon\omega=0$ of the
integrable foliation given by $F$.
  We study different invariants: the orbit depth $k$, the nilpotence class}$n$,
the derivative length}$d$ associated with the couple $(F,\gamma)$. These
invariants bound the length $\ell$ of the first nonzero Melnikov function of
the deformation $dF+\epsilon\omega$ along $\gamma$.
  We study in detail a simple example of a polynomial $F$ given as product of
four lines. We show how these invariants vary depending on the relative
position of the four lines and relate it also to the length of the
corresponding Godbillon-Vey sequence. We formulate a conjecture motivated by
the study of this example.
