The inverse problem of differential Galois theory over the field R(z)

We describe a Picard-Vessiot theory for differential fields with non algebraically closed fields of constants. As a technique for constructing and classifying Picard-Vessiot extensions, we develop a Galois descent theory. We utilize this theory to prove that every linear algebraic group $G$ over $\mathbb{ R}$ occurs as a differential Galois group over $\mathbb{ R}(z)$. The main ingredient of the proof is the Riemann-Hilbert correspondence for regular singular differential equations over $\mathbb{ C}(z)$.

Categories: math.CA math.AG