ON A GENERIC INVERSE DIFFERENTIAL GALOIS PROBLEM FOR GL_{n}

LOURDES JUAN

Abstract. Let F be a differential field with algebraically closed field of constants C. We prove that F ⟨Y_{ij}⟩(X_{ij}) ⊃ F ⟨Y_{ij}⟩ is a generic Picard-Vessiot extension of F for GL_{n}(C). If E ⊃ F is any Picard-Vessiot extension with differential Galois group GL_{n}(C) then E ≅ F (X_{ij}) as F - and GL_{n}(C)-modules and there are f_{ij} ∈ F such that F ⟨Y_{ij}⟩(X_{ij}) ⊃ F ⟨Y_{ij}⟩ specializes to E ⊃ F via Y_{ij} ↦ f_{ij}. The [f_{ij}] ∈ M_{n}(F) for which the image of the map Y_{ij} ↦ f_{ij} is a Picard-Vessiot extension of F with group GL_{n}(C) can be characterized as those [f_{ij}] ∈ M_{n}(F) for which the wronskians of the monomials in F ⟨Y_{ij}⟩(X_{ij}) of degree less than or equal to k all map to non-zero elements under Y_{ij} ↦ f_{ij}.