Connected Linear Groups as Differential Galois Groups

Abstract

THEOREM 1.1. Let C be an algebraically closed field of characteristic zero, G a connected linear algebraic group defined o er C, and k a differential field containing C as its field of constants and of finite, nonzero transcendence degree o er C; then G can be realized as the Galois group of a Picard]Vessiot extension of k. w x Previous work by Kovacic 17, 18 reduced this problem to the case of powers of a simple connected linear algebraic group. Our contribution is to show that one is able to realize any connected semisimple group as a Ž . Galois group and, when k s C x , x9 s 1, to control the number and types of singularities when one constructs a system Y 9 s AY, A g Ž Ž .. M C x , realizing an arbitrary connected linear algebraic group as its n