In several classical families of differential equations such as the Painlevé families (Nagloo, Pillay) or finite dimensional families of Schwarzian differential equations (Blazquez-Sanz, Casale, Freitag, Nagloo), the following picture has been obtained regarding the transcendence properties of their solutions:
– (Strong minimality): outside of an exceptional set of parameters, the corresponding differential equations are strongly minimal,
– (Geometric triviality): algebraic independence of several solutions is controlled by pairwise algebraic independence outside of this exceptional set of parameters,
– (Multidimensionality): the differential equations defined by generic independent parameters are orthogonal.
Are the families of differential equations satisfying such transcendence properties scarce or abundant in the universe of algebraic differential equations?
I will describe an abundance result for families of autonomous differential equations satisfying the first two properties. The model-theoretic side of the proof uses a fine understanding of the structure of autonomous differential equations internal to the constants that we have recently obtained in a joint work with Rahim Moosa. The geometric side of the proof uses a series of papers of S.C. Coutinho and J.V. Pereira on the dynamical properties of a generic foliation.