Submission date: 7 February 2017

Abstract:

In an earlier paper it was proved that if a differential field (K,δ) 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.

Mathematics Subject Classification: 12H05 03C60

Keywords and phrases: Picard-Vessiot extensions, differential Galois cohomology

Full text arXiv 1702.01969: pdf, ps.