In this paper we solve some problems posed by Kolchin about differential algebraic groups. The main result (from which the others follow) is the embeddability of any differential algebraic group in an algebraic group. A crucial intermediate result, and one of independent interest, is a generalisation of Weil’s theorem on recovering an algebraic group from birational data, to pro-algebraic groups.

/pdf/some-foundational-questions-concerning-differential-1effl4al6a.pdf

Some foundational questions concerning differential algebraic groups

This paper continues previous work in which we developed the Galois theory of strongly normal extensions using differential schemes. In the present paper we derive two main results. First, we show that an extension is strongly normal if and only if a certain differential scheme splits, i.e. is obtained by base extension of a scheme over constants. This gives a geometric characterization to the notion of strongly normal. Second, we show that Picard-Vessiot extensions are characterized by their Galois group being affine. Our proofs are elementary and do not use "group chunks" or cohomology. We end by recalling some important results about strongly normal extensions with the hope of spurring future research.

/pdf/geometric-characterization-of-strongly-normal-extensions-2untmc5dlp.pdf

Geometric characterization of strongly normal extensions

The Picard–Vessiot Antiderivative Closure

In a survey the main results in differential and difference algebra (differential rings and modules; differential fields; differential-algebraic geometry; differential and difference dimension; differential and difference Galois theory; integration in finite terms; differential algebraic groups), compiled from the reviews published inReferativnyl Zhurnal Matematika for the years 1975–1986, are reflected.

Differential and difference algebra

We revisit Kolchin's results on definability of differential Galois groups of strongly normal extensions, in the case where the field of constants is not necessarily algebraically closed. In certain classes of differential topological fields, which encompasses ordered or p-valued differential fields, we find a partial Galois correspondence and we show one cannot expect more in general. In the class of ordered differential fields, using elimination of imaginaries in CODF, we establish a relative Galois correspondence for relatively definable subgroups of the group of differential order automorphisms.

On differential Galois groups of strongly normal extensions

Pro-Algebraic Groups and the Galois Theory of Differential Fields

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Pro-Algebraic Groups and the Galois Theory of Differential Fields