Cyclicity of some symmetric nilpotent centers

Abstract

In this work we present techniques for bounding the cyclicity of a wide class of monodromic nilpotent singularities of symmetric polynomial planar vector fields. The starting point is identifying a broad family of nilpotent symmetric fields for which existence of a center is equivalent to existence of a local analytic first integral, which, unlike the degenerate case, is not true in general for nilpotent singularities. We are able to relate so-called “focus quantities” to the “Poincaré–Lyapunov quantities” arising from the Poincaré first return map. When we apply the method to concrete examples, we show in some cases that the upper bound is sharp. Our approach is based on computational algebra methods for determining a minimal basis (constructed by focus quantities instead of by Poincaré–Lyapunov quantities because of the easier computability of the former) of the associated polynomial Bautin ideal in the parameter space of the family. The case in which the Bautin ideal is not radical is also treated.