On a generic inverse differential Galois problem for $\rm GL_n$
Lourdes Juan
Let $F$ be a differential field with algebraically closed field of constants $C$. We prove that $F\langle Y_{ij}\rangle( X_{ij})\supset F\langle Y_{ij}\rangle$ is a generic Picard-Vessiot extension of $F$ for ${{\rm {GL}_n}(C)}$. If $E\supset F$ is any Picard-Vessiot extension with differential Galois group ${{\rm {GL}_n}(C)}$ then $E\cong F( X_{ij})$ as $F$- and ${{\rm {GL}_n}(C)}$-modules and there are $f_{ij}\in F$ such that $F\langle Y_{ij}\rangle( X_{ij})\supset F\langle Y_{ij}\rangle$ 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 ${{\rm {GL}_n}(C)}$ can be characterized as those $[f_{ij}]\in M_n(F)$ for which the Wronskians of the monomials in $F\langle Y_{ij}\rangle( X_{ij})$ of degree less than or equal to $k$ all map to non-zero elements under $ Y_{ij} \mapsto f_{ij}$.