The differential Galois group of the rational function field

Michael Wibmer; Julia Hartmann; David Harbater; Annette  Bachmayr

doi:10.1016/j.aim.2021.107605

The differential Galois group of the rational function field

Michael Wibmer, Julia Hartmann, David Harbater, Annette Bachmayr

Institute of Analysis and Number Theory (5010)

Research output: Contribution to journal › Article › peer-review

Abstract

We determine the absolute differential Galois group of the field C(x) of rational functions: It is the free proalgebraic group on a set of cardinality |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 C(x) also applies to rational function fields over more general fields of coefficients.

Original language	English
Article number	107605
Journal	Advances in Mathematics
Volume	381
DOIs	https://doi.org/10.1016/j.aim.2021.107605
Publication status	Published - 2021

Keywords

Differential algebra
Embedding problems
Inverse differential Galois problem
Linear algebraic groups
Picard-Vessiot theory
Proalgebraic groups

ASJC Scopus subject areas

General Mathematics

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10.1016/j.aim.2021.107605

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title = "The differential Galois group of the rational function field",

abstract = "We determine the absolute differential Galois group of the field C(x) of rational functions: It is the free proalgebraic group on a set of cardinality |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 C(x) also applies to rational function fields over more general fields of coefficients.",

keywords = "Differential algebra, Embedding problems, Inverse differential Galois problem, Linear algebraic groups, Picard-Vessiot theory, Proalgebraic groups",

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AU - Wibmer, Michael

AU - Hartmann, Julia

AU - Harbater, David

AU - Bachmayr, Annette

PY - 2021

Y1 - 2021

N2 - We determine the absolute differential Galois group of the field C(x) of rational functions: It is the free proalgebraic group on a set of cardinality |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 C(x) also applies to rational function fields over more general fields of coefficients.

AB - We determine the absolute differential Galois group of the field C(x) of rational functions: It is the free proalgebraic group on a set of cardinality |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 C(x) also applies to rational function fields over more general fields of coefficients.

KW - Differential algebra

KW - Embedding problems

KW - Inverse differential Galois problem

KW - Linear algebraic groups

KW - Picard-Vessiot theory

KW - Proalgebraic groups

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VL - 381

JO - Advances in Mathematics

JF - Advances in Mathematics

M1 - 107605

ER -

The differential Galois group of the rational function field

Abstract

Keywords

ASJC Scopus subject areas

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