On a generic inverse differential Galois problem for GL_n

Abstract

\newcommand{\GLn}{\operatorname{GL}_n} \newcommand{\GL}{\GLn(C)} Let $F$ be a differential field with algebraically closed field of constants $C$. We prove that $F<Y_{ij}>(X_{ij})\supset F<Y_{ij}>$ is a generic Picard-Vessiot extension of $F$ for $\GL$. If $E\supset F$ is any Picard-Vessiot extension with differential Galois group $\GL$ then $E\cong F(X_{ij})$ as $F$- and $\GL$-modules and there are $f_{ij}\in F$ such that $F<Y_{ij}>(X_{ij})\supset F<Y_{ij}>$ specializes to $E\supset F$ via $Y_{ij}\mapsto f_{ij}$. The $[f_{ij}]\in M_n(F)$ for which the image of the map $Y_{ij}\mapsto f_{ij}$ is a Picard-Vessiot extension of $F$ with group $\GL$ can be characterized as those $[f_{ij}]\in M_n(F)$ 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}\mapstof_{ij}$.