Interpretations and differential Galois extensions

We give accounts and proofs, using model-theoretic methods among other things, of the following results: Suppose ∂y = Ay is a linear differential equation over a differential field K of characteristic 0, and the field CK 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 L1 and L2 are two Picard–Vessiot extensions of K which (as fields) have a common embedding over K into an elementary extension of CK, then L1 and L2 are isomorphic over K as differential fields; and (iii) suppose that CK 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 CK 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 CK, 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 H1(k,G) is countable. The current paper replaces the preprint [8] which only dealt with the linear differential equations case and had some mistakes.