Interpretations and Differential Galois Extensions

We give accounts and proofs, using model-theoretic methods among other things, of the following results: Suppose partial derivative y = Ay is a linear differential equation over a differential field K of characteristic 0, and the field C-K of constants of K is existentially closed in K. Then, (i) there exists a Picard-Vessiot extension L of K, namely a differential field extension L of K which is generated by a fundamental system of solutions of the equation, and has no new constants; (ii) if L-1 and L-2 are two Picard-Vessiot extensions of K which (as fields) have a common embedding over K into an elementary extension of C-K, then L-1 and L-2 are isomorphic over K as differential fields; and (iii) suppose that C-K is large in the sense of Pop [21] and also has only finitely many extensions of degree n for all n (Serre's property (F)). Then, K has a Picard-Vessiot extension L such that C-K is existentially closed in L. In fact we state and prove our results in the more general context of logarithmic differential equations over K on (not necessarily linear) algebraic groups over C-K, and the corresponding strongly normal extensions of K. We make use of interpretations from model theory as well the Galois groupoid, which are related to the Tannakian theory in [3, 4], but go beyond the linear context. Towards the proof of (iii) we obtain a Galois-cohomological result of possibly independent interest: if k is a field of characteristic 0 with property (F), and G is any algebraic group over k, then H-1(k, G) is countable. The current paper replaces the preprint [8] which only dealt with the linear differential equations case and had some mistakes.