Godbillon-Vey sequence and Francoise algorithm

We consider foliations given by deformations dF + epsilon omega of exact forms dF in C-2 in a neighborhood of a family of cycles -gamma(t) subset of F-1(t). In 1996 Francoise gave an algorithm for calculating the first nonzero term of the displacement function Delta along gamma of such deformations. This algorithm recalls the well-known Godbillon-Vey sequences discovered in 1971 for investigation of integrability of a form omega. In this paper, we establish the correspondence between the two approaches and translate some results by Casale relating types of integrability for finite Godbillon-Vey sequences to the Francoise algorithm settings. (C) 2019 Published by Elsevier Masson SAS.