Résume | In the Galois theory of linear differential equations, the Picard--Vessiot extensions are differential field extensions that are analogous to the splitting fields in usual Galois theory. When the field of constants is algebraically closed, a classical result asserts that such an extension exists for every linear equation, and it is unique up to isomorphism. However, examples show that this fails when the constants are not algebraically closed.
I will discuss a joint work with A. Pillay, where we show that Picard-Vessiot (and more generally, strongly normal) extensions exist whenever the field of constants is existentially closed (as a field) in the base field. Furthermore, with some additional field-theoretic assumptions, we obtain that the field of constants is existentially closed in the extension, and also a suitable uniqueness result. This generalises results of Crespo-Hajto-van der Put and others. |