THE NILPOTENT PART AND DISTINGUISHED FORM OF RESONANT VECTOR-FIELDS OR DIFFEOMORPHISMS

We investigate two important objects attached to (local, analytic, and above all resonant) vector fields or diffeomorphisms, namely the nilpotent part (a classical, almost self-explanatory notion, but with a hitherto poorly understood pattern of divergence) and the distinguished form (a spherical ''resonant form'', chart-dependent, yet parfectly canonical). We find their analytical expression in terms of two universal sets of coefficients (the moulds $. and $.) and completely describe their divergence properties with the help of resurgence equations. We derive therefrom yet another means of calculating all the holomorphic invariants of ''local objects'' (fields or diffeos), and conclude by sketching a few algebraic consequences of our construction.