Amador Martin-Pizarro — Model theory, differential algebra and functional transcendence , after Freitag, Jaoui, and Moosa

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 and Jaoui together with Marker and Nagloo established transcendence results for the solutions of differential equations of Liénard type, generalizing existing work by Poizat. They noticed that, given any finite collection of pairwise distinct (non-trivial) solutions to the Poizat equation, a differential equation of Liénard type over the constants of order 2, the solutions and their first derivatives are algebraically independent, that is, they satisfy no non-trivial algebraic relation. There are two steps for this: for Poizat’s equation, an algebraic dependence between finitely many solutions and their first derivatives arises from a certain algebraic dependence between two of them. Now, Poizat’s equation has the D₂ property, meaning that given two distinct solutions, there is no non-trivial algebraic dependence between the solutions and their first derivatives.

Poizat’s equation is actually strongly minimal, that is, irreducible of (Morley) rank 1, a fundamental notion in model theory which allows model theorists to analyse the geometric behaviour of algebraic differential equations in terms of the strongly minimal ones as the building blocks. The trichotomy theorem of Hrushovski and Sokolović for differentialy closed fields of characteristic 0 implies that, if a strongly minimal differential equation of order n defined over the constants has property D₂, then any finite set collection of pairwise distinct solutions together with their derivatives up to order n-1 are algebraically independent. Not every strongly minimal differential equation defined over the constants has property D₂. For example, the j-invariant function, which encodes the isomorphism classes of elliptic curves over the complex numbers, is a solution of a differential equation defined over the constants expressed in terms of the schwarzian derivative. Freitag and Scanlon showed that this equation is strongly minimal, using Pila’s modular Ax-Schanuel, yet it does not have property D₂, witnessed indeed by the modular relations given by Hecke correspondences.

Now, showing strong minimality for a differential equation is far from obvious, for it already requires a good understanding of the relations between its solutions. Freitag, Jaoui and Moosa have generalised in recent work the above for any differential equation given by an irreducible differential polynomial of order n defined over the constants, regardless of whether the equation is strongly minimal. That is, they show that, if the equation has property D₂, any finite collection of pairwise distinct solutions together with their derivatives up to order n-1 are algebraically independent. Their proof 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 the proof of Freitag, Jaoui and Moosa, without assuming a deep knowledge in (geometric) model theory (but some familiarity with basic notions in algebraic geometry).