Appears in collection : Bourbaki - Novembre 2025

By Amador MARTIN-PIZARRO Abstract : A fundamental problem in the study of algebraic differential equations is determining the possible algebraic relations among different solutions of a given differential equation. Freitag, Jaoui, and Moosa have isolated an essential property, called property D2, in or- der to show that if a differential equation given by an irreducible differential polynomial of order n is defined over the constants and has property D2, then any number of pairwise distinct solutions together with their derivatives up to order n−1 are algebraically inde- pendent. The property D2 requires that, given two distinct solutions, there is no non-trivial algebraic dependence between the solutions and their first n−1 derivatives. The proof of Freitag, Jaoui and Moosa is extremely elegant and short, yet it uses in a clever way fundamental results of the model theory of differentially closed fields of characteristic 0. The goal of this talk is to introduce the model-theoretic tools at the core of their proof, without assuming a deep knowledge in (geometric) model theory (but some familiarity with basic notions in algebraic geometry).