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arXiv · 2004.05906

The differential Galois group of the rational function field

Annette Bachmayr·David Harbater·Julia Hartmann·Michael Wibmer
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Abstract

We determine the absolute differential Galois group of the field $\mathbb{C}(x)$ of rational functions: It is the free proalgebraic group on a set of cardinality $|\mathbb{C}|$. This solves a longstanding open problem posed by B.H. Matzat. For the proof we develop a new characterization of free proalgebraic groups in terms of split embedding problems, and we use patching techniques in order to solve a very general class of differential embedding problems. Our result about $\mathbb{C}(x)$ also applies to rational function fields over more general fields of coefficients.

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Annette Bachmayr, David Harbater, Julia Hartmann, Michael Wibmer. 2020-04-13. The differential Galois group of the rational function field. https://arxiv.org/abs/2004.05906

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