Interpretations and differential Galois extensions. DRAFT

We give model-theoretic accounts and proofs 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) There is a field C of constants which is an elementary extension of CK such that K(C) has field of constants C, and has a Picard-Vessiot extension L such that CK is existentially closed in L. (iiii) Assume that the field CK has finitely many extensions of degree n for all n. Then in (ii) we can choose C to be CK , namely already K has a Picard Vessiot extension L such that CK is existentially closed in L. (iv) 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. Our results are proved in the more general context of logarithmic differential equations over K on not necessarily linear algebraic groups ∗Partial support from MSRI and NSF