Cyclicity of a fake saddle inside the quadratic vector fields

Agraïments: The first author acknowledges the support of FWO Vlaanderen (Grant number G093910). UNAB10-4E-378 grantsThis paper concerns the study of small-amplitude limit cycles that appear in the phase portrait near an unfolded fake saddle singularity. This degenerate singularity is also known as a impassable grain. The normal form of the unperturbed vector field is like a degenerate flow box. Near the singularity,the phase portrait consists of parallel fibers, all of which but one have no singular points, and at the singular fiber, there is one node. We study different techniques in order to show that the cyclicity is bigger or equal than two when the normal form is quadratic