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

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

2008-02-20

arxiv.org/abs/0802.2897v1

Mathematics

Classical Analysis and ODEs

23 pages

Scientific paper

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 $\mathbb{C}(z)$.

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