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.
Highlights
This paper deals mainly with differential fields (K, ∆) of characteristic 0, where ∆ = {δ1, . . . , δm} is a set of commuting derivations on K
In the case of m > 1 condition (1) will be replaced by “K is algebraically closed and has no proper generalized Picard–Vessiot extensions”. Even with this rather inclusive condition, the equivalence with (2) will fail, basically due to the existence of proper definable subgroups of the additive group which are orthogonal to all proper definable subfields
The problem arises from the existence of proper definable subgroups of the multiplicative group which are orthogonal to the fixed field
Summary
In the case of m > 1 condition (1) will be replaced by “K is algebraically closed and has no proper generalized Picard–Vessiot extensions” ( something slightly stronger). Even with this rather inclusive condition, the equivalence with (2) will fail, basically due to the existence of proper definable subgroups of the additive group which are orthogonal to all proper definable subfields. The problem arises from the existence of proper definable subgroups of the multiplicative group which are orthogonal to the fixed field
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