An improved version of Poincare-Dulac theorem for improved nilpotent normal forms

An improved version of the well-known Poincare-Dulac’s normal form theoremis first proposed. It is shown that, for a nonlinear vector field, a normal formnear a singular point can always be chosen so that the number of nonlinearcomponents is at most equal to the number of Jordan blocks in the normalizedleading matrix, thus leading to the “simplest” form in which a formal vectorfield can be written near a singular point. Within this scope, a generalizationto any dimension of an important result on normal forms of nilpotent systemsis given. This is the main result of the paper.