Équation homologique et cycles asymptotiques d'une singularité nœud-col

Let a germ of holomorphic vector field Z be given and assume that is an isolated degenerate-resonnant singular point for Z (one and only one non-zero eigenvalue). Such a vector field acts as a derivative over the space of holomorphic germs at the origin of the complex plane. We obtain the solutions of the homological equation Z·F=G by integrating G along some asymptotic paths tangent to the complex trajectories of Z and ending at the singularity; this locate the obstructions to solve such an equation in the period of G along asymptotic cycles. The Borel transform is thus extended to the foliated setting and this geometrical approach helps us in the study of the conjugacy problem. For instance we find without expense of computation the obstructions obtained previously by P.M. Elizarov for the Poincaré-Dulac models. This approach of the caracteristics method in the singular setting will lead us, in a further print, to describe the analytical classification of germs of degenerate-resonnant vector fields.