Splitting differential algebraic groups (Q2639104)

D is a finite set of commuting derivations of a universal characteristic zero differential field U with field of constants K and F is a differential subfield of U with field of constants C. A linear D-F-group is a D-F-closed subgroup G of some \(Gl_ n(U)\); it is split if G is the intersection of some C-closed subgroup of \(Gl_ n(U)\) with \(Gl_ n(K)\); it is splittable if it is split over some extension of F. The author's main result is that an irreducible D-F-group G whose function field has finite transcendence degree over F and whose radical is unipotent, is splittable over some Picard-Vessiot extension of F. He begins by showing that under the transcendence degree assumption the D-F- coordinate ring \(F\{\) \(G\}\) of G is finitely generated as an F-algebra. After observing that finite-dimensional D-F-vector spaces are splittable over Picard-Vessiot extensions, he reduces the problem to showing that \(F\{\) \(G\}\) is locally finite as a D-F-vector space. Then the author uses the unipotent radical property to obtain this latter property and hence his main result.