2911 20170113100534.0 SCART-2016-0011 Stolovitch, L Holomorphic Normal Form of Nonlinear Perturbations of Nilpotent Vector Fields 2016 We consider germs of holomorphic vector fields at a fixed point having a nilpotent linear part at that point, in dimension 3. Based on Belitskii’s work, we know that such a vector field is formally conjugate to a (formal) normal form. We give a condition on that normal form which ensures that the normalizing transformation is holomorphic at the fixed point. We shall show that this sufficient condition is a nilpotent version of Bruno’s condition (A). In dimension 2, no condition is required since, according to Strozyna – Zoladek, each such germ is holomorphically conjugate to a Takens normal form. Our proof is based on Newton’s method and sl2(C)-representations. NO local analytic dynamics, fixed point, normal form, Belitskii normal form, small divisors, Newton method, analytic invariant manifold, complete integrability Verstringe, F Regular and chaotic dynamics 21 2016 4 410-436 freek.verstringe@observatoire.be http://arxiv.org/pdf/1606.07242.pdf accepted to be published in REFERD