The complex viewpoint for transverse impasse points of quasi-linear differential equations

Abstract

The main purpose of this paper is to show that the singularities called impasse points of real implicit ordinary differential equations are generically branch points of the solution, using a complex viewpoint. This research starts from a result asserting that the real solution in most cases behaves like ±x−x0 at an impasse point (x0,y0). We extend the notion of regular impasse point defined in previous works, and consider instead transverse impasse point, where the underlying vector field is transverse to the singular locus, even in the case when this hypersurface is not a manifold locally.Here we make use of Puiseux expansions and we show that, under generic hypotheses, the solution is multivalued at such a point. We prove that the Puiseux exponent is related simply to the multiplicity of the impasse point in the singular locus: if M is the total multiplicity of the singularity (z0,y0), there is a unique solution at this point and it is M+1-valued.