We present the main ideas for proving o-minimality of the expansion of the real field generated by all Dulac transition maps near a non-resonant hyperbolic singularity of a planar analytic vector field. We also show how the existence of (non-explicit) uniform bounds on the number of limit cycles of certain (very special) families of analytic vector fields can be obtained from our approach.

Jean-Philippe Rolin, Dmitry Novikov, Sergei Yakovenko