A generic Picard-Vessiot extension for GL(n) and the inverse differential Galois problem.

Abstract

Let F be a differential field with field of constants the algebraically dosed field C. Let Yij be differential indeterminates and let R = F{ Yij}[X11, ..., Xnn] be a differential ring with derivation D (Xij) = k=1n YikXkj. We show that the quotient field Q = F⟨ Yij⟩ ( Xij) of R is a Picard-Vessiot extension of F(Yij) for GL n(C). Moreover, we show that Q is a generic extension in the following sense: If E ⊃ F is a Picard-Vessiot extension with differential Galois group G = GLn( C) then E is isomorphic to F( Xij) as a G-module and as an F-module. Under this isomorphism DE, the derivation of E, goes to a G-equivariant derivation which has a form similar to D with the coefficients Yij specialized to elements fij from F. The differential subfield C⟨ fij⟩ (Xij) ⊂ F(Xij) is shown to be a Picard-Vessiot extension of C⟨ fij⟩ with group GLn(C). From this, one can retrieve the Picard-Vessiot extension F( Xij) ⊃ F by extension of scalars from C⟨ fij⟩ to F.

Conversely, if given F we specialize the Y ij to fij in F so that the corresponding extension C⟨ f ij⟩ (X) ⊃ C⟨ fij⟩ has no new constants we obtain a solution to the inverse differential Galois problem for GLn(C). In the second part of this dissertation we show necessary and sufficient conditions for such a specialization to exist.