Apparaît dans la collection : Bourbaki - Novembre 2025

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 $D_2$, in order to show that if a differential equation given by an irreducible differential polynomial of order n is defined over the constants and has property $D_2$, then any number of pairwise distinct solutions together with their derivatives up to order $n-1$ are algebraically independent. The property $D_2$ 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).

[After Freitag, Jaoui, and Moosa]