Picard-Vessiot extensions, linear differential algebraic groups and their torsors

TL;DR

This paper explores the structure of Picard-Vessiot extensions and differential algebraic groups over differential fields, establishing density results and connections between torsors and extensions, with implications for differential field boundedness.

Contribution

It proves density of rational points in differential algebraic torsors over Picard-Vessiot closures and links torsors to Picard-Vessiot extensions, extending the theory of differential algebraic groups.

Findings

X(L) is Kolchin-dense in X for differential torsors.

When G is finite-dimensional, X(L) equals X(K^diff).

Relationships between Picard-Vessiot extensions and torsors are established.

Abstract

Let K be differential field with algebraically closed field of constants. Let K^diff be a differential closure of K, and L the (iterated) Picard-Vessiot closure of K inside K^diff. Let G be a linear differential algebraic group over K and X a differential algebraic torsor for G over K. We prove that X(L) is Kolchin-dense in X. When G is finite-dimensional we prove that X(L) = X(K^diff). We give close relationships between Picard-Vessiot extensions of K and torsors for suitable finite-dimensional linear differential algebraic groups over K. We suggest some differential field analogues of the notion of boundedness for fields (Serre's property F).