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Connected Linear Groups as Differential Galois Groups over $C(x)$.
Claude Mitschi1, Michael F. Singer

We generalize results of Kovacic to solve the inverse problem in differential Galois theory for connected linear groups, over $C(x)$ where $C$ is an arbitrary algebraically closed field $C$ of characteristic zero. In particular we show: {it Let $G$ be a connected linear algebraic group defined over an algebraically closed field $C$ of characteristic zero. Then G is the Galois group of a Picard-Vessiot extension of $C(x)$ corresponding to a system of the form $ Y' = (frac{A_1}{x-alpha_1} + ldots + frac{A_{d(G)}}{x-alpha_{d(G)}} + A_{d(G)+1}) Y$ where $A_1, ldots, A_{d(G)}$ are constant matrices and $A_{d(G)+1}$ is a matrix with polynomial entries of degree at most $e(G)$. In particular, this system has $d(G)$ regular singular points in the finite plane and a (possibly) irregular singular point at infinity.} Here $d(G)$ is defined to be the dimension of $R_u/(G,R_u)$ where $R_u$ is the unipotent radical of $G$ and $(G,R_u)$ is the commutator group of $G$ and $R_u$. The bound $e(G)$ is defined in a similar way from the adjoint action of a Levi factor on $R_u$. More generally, using a lemma of Kovacic, we get the following result: {it Let $C$ be an algebraically closed field of characteristic zero, $G$ a connected linear algebraic group defined over $C$ and $k$ a differential field containing $C$ as its field of constants and of finite, nonzero transcendence degree over $C$, then $G$ can be realized as the Galois group of a Picard-Vessiot extension of $k$. } Our proof is constructive and purely algebraic. It not only solves the inverse problem but reproves and generalizes a result of Ramis that states that if $G$ is a connected linear algebraic group with $d(G)=1$, then $G$ is the Galois group of a system having at most two singular points, one regular and one irregular. Ramis also conjectured that when $d(G) = 0$, one singularity suffices. Our result confirms this conjecture as well.
1 :  IRMA - Institut de Recherche Mathématique Avancée
"Linear algebraic groups – Differential Galois groups – Inverse Galois problems – Lie algebras"