Generalized Picard-Vessiot extensions and differential Galois cohomology

Abstract

In an earlier paper it was proved that if a differential field $(K,\delta )$ is algebraically closed and closed under Picard-Vessiot extensions then every differential algebraic principal homogeneous space over K for a linear differential algebraic group G over K has a K-rational point (in fact if and only if). This paper explores whether and if so, how, this can be extended to (a) several derivations, (b) one automorphism. Under a natural notion of "generalized Picard-Vessiot extension" (in the case of several derivations), we give a counterexample. We also have a counterexample in the case of one automorphism. We also formulate and prove some positive statements in the case of several derivations.

arXiv commentThe new version is the one accepted by Annales Math. Toulouse, and makes some clarifications and additions suggested by the referees, including reference to a recent related paper by Minchenko and Ovchinnikov