A preparation theorem for codimension-one foliations

Abstract

After gluing foliated complex manifolds, we derive a preparation-like theorem for singularities of codimension-one foliations and planar vector fields (in the real or complex setting). Without computation, we retrieve and improve results of Levinson-Moser for functions, Dufour-Zhitomirskii for nondegenerate codimension-one foliations (proving in turn the analyticity), Stróżyna-Żoładek for non degenerate planar vector fields and Bruno-Écalle for saddle-node foliations in the plane.