Nilpotence of Orbits under Monodromy and the Length of Melnikov Functions

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.