Model completion of Lie differential fields

Abstract

We define a Lie differential field as a field $\text{A}$ of characteristic 0 with an action, as derivations on $\text{A}$, of some given Lie algebra $\text{L}$. We assume that $\text{L}$ is a finite-dimensional vector space over some sub-field $\text{F}\text{⊆}\text{A}$ given in advance. As an example take the field of rational functions on a smooth algebraic variety, with $\text{L}\text{={}\text{rational}\text{vector}\text{fields}\text{}}$.

For every simple extension of Lie differential fields we find a finite system of differential equations that characterizes it. We then define, using first-order conditions, a collection of allowed systems of differential equations s.t. the above characteristic systems are allowed. We prove that for every allowed system there exists a generic solution in some extension, and this solution is unique (up to isomorphism).

We construct the model completion of the theory of Lie differential fields by adding axioms stating that every allowed system has almost generic solutions. The construction is a generalization of Blum's axioms for $\text{DCF}{}_0$. We also show that this model completion is $\text{ω}$-stable.