Model completion of Lie differential fields Original Research Article

We define a Lie differential field as a field View the MathML source of characteristic 0 with an action, as derivations on View the MathML source, of some given Lie algebra View the MathML source. We assume that View the MathML source is a finite-dimensional vector space over some sub-field View the MathML source given in advance. As an example take the field of rational functions on a smooth algebraic variety, with View the MathML source. 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 Blumʹs axioms for View the MathML source. We also show that this model completion is ω-stable.