Analytic continuation and fixed points of the Poincaré mapping for a polynomial Abel equation

International audienceWe consider an Abel differential equation y′=p(x)y2+q(x)y3 with p(x), q(x) polynomials in x. For two given points a and b in C, the ``Poincar\'e mapping" of the above equation transforms the values of its solutions at a into their values at b. In this paper we study global analytic properties of the Poincar\'e mapping, in particular, its analytic continuation, its singularities and its fixed points (which correspond to the ``periodic solutions" such that y(a)=y(b)). On the one hand, we give a general description of singularities of the Poincar\'e mapping, and of its analytic continuation. On the other hand, we study in detail the structure of the Poincar\'e mapping for a local model near a simple fixed singularity, where an explicit solution can be written. Yet, the global analytic structure (in particular, the ramification) of the solutions and of the Poincar\'e mapping in this case is fairly complicated, and, in our view, highly instructive. For a given degree of the coefficients we produce examples with an infinite number of {\it complex} ``periodic solutions" and analyze their mutual position and branching. Let us recall that Pugh's problem, which is closely related to the classical Hilbert's 16th problem, asks for the existence of a bound to the number of {\it real} isolated ``periodic solutions". New findings reported here lead us to propose new insights on the Poincar\'e mapping. If the ``complexity" of the path in the x-plane between a and b is {\it a priori} bounded, the number of fixed points should be uniformly bounded. We think that, in some sense, this is close to the complex version of Khovansky's fewnomial theory