Perturbative Methods for the Painlevé Test

There exist many situations where an ordinary differential equation admits a movable critical singularity which the test of Kowalevski and Gambier fails to detect. Some possible reasons are : existence of negative Fuchs indices, insufficient number of Fuchs indices, multiple family, absence of an algebraic leading order. Mainly giving examples, we present the methods which answer all these questions. They are all based on the theorem of perturbations of Poincaré and computerizable.