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

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 $\mathbb{C}(z)$.

arXiv comment23 pages

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• Existence of $\partial$-parameterized Picard-Vessiot extensions over fields with algebraically closed constants
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• Root Parametrized Differential Equations for the classical groups
• Picard-Vessiot Extensions of Real Differential Fields
• Non-free iterative differential modules