Let $F\in ℂ[x,y]$ be a polynomial, $\gamma \left(z\right)\in {\pi }_1\left(F^{-1}\left(z\right)\right)$ 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 bind the length $\ell$ of the first nonzero Melnikov function of the deformation $dF+\epsilon \omega$ along $\gamma$. We analyze the variation of the aforementioned invariants in a simple but informative example, in which the polynomial $F$ is defined by a product of four lines. We study as well the relation of this behavior with the length of the corresponding Godbillon–Vey sequence. We formulate a conjecture motivated by the study of this example.

让 $F\in ℂ[X,是]$ 是多项式， $\gamma \left(z\right)\in {\pi }_1\left(F^{-1}\left(z\right)\right)$ 通用纤维中的非平凡循环 $F$ 然后让 $\omega$ 是多项式 1-形式，从而定义多项式变形 $dF+\epsilon \omega =0$ 的可积叶理由下式给出 $F$.

我们研究不同的不变量：轨道深度 $克$，幂零类 $n$,导数长度 $d$ 与这对夫妇有关 $(F,\gamma )$. 这些不变量绑定了长度 $\ell$ 变形的第一个非零 Melnikov 函数$dF+\epsilon \omega$ 沿着 $\gamma$. 我们在一个简单但信息丰富的例子中分析了上述不变量的变化，其中多项式$F$由四行的乘积定义。我们还研究了这种行为与相应的 Godbillon-Vey 序列长度的关系。我们通过对这个例子的研究提出了一个猜想。