6533b824fe1ef96bd1280c47
RESEARCH PRODUCT
The differential Galois group of the rational function field
Julia HartmannAnnette BachmayrMichael WibmerDavid Harbatersubject
Pure mathematicsGroup (mathematics)General Mathematics010102 general mathematicsGalois groupField (mathematics)Rational functionMathematics - Commutative AlgebraCommutative Algebra (math.AC)01 natural sciences12H05 12F12 34M50 14L15Mathematics - Algebraic Geometry0103 physical sciencesFOS: MathematicsEmbeddingOrder (group theory)Differential algebra010307 mathematical physics0101 mathematicsAlgebraic Geometry (math.AG)Picard–Vessiot theoryMathematicsdescription
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.
| year | journal | country | edition | language |
|---|---|---|---|---|
| 2020-04-13 |