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

Tobias Dyckerhoff

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arXiv · 0802.2897

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

Tobias Dyckerhoff

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Abstract

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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Tobias Dyckerhoff. 2008-02-20. The inverse problem of differential Galois theory over the field R(z). https://arxiv.org/abs/0802.2897

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