Nilpotence of orbits under monodromy and the length of Melnikov functions

Let F is an element of C[x, y] be a polynomial, gamma(z) is an element of 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 + c 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 l of the first nonzero Melnikov function of the deformation dF+ epsilon omega along.. 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. (C) 2021 Elsevier B.V. All rights reserved.