Search results for "12H10"

showing 2 items of 2 documents

Algebraic groups as difference Galois groups of linear differential equations

2019

We study the inverse problem in the difference Galois theory of linear differential equations over the difference-differential field $\mathbb{C}(x)$ with derivation $\frac{d}{dx}$ and endomorphism $f(x)\mapsto f(x+1)$. Our main result is that every linear algebraic group, considered as a difference algebraic group, occurs as the difference Galois group of some linear differential equation over $\mathbb{C}(x)$.

Linear algebraic groupPure mathematicsAlgebra and Number TheoryEndomorphism010102 general mathematicsGalois theoryGalois groupField (mathematics)Commutative Algebra (math.AC)Mathematics - Commutative Algebra01 natural sciencesMathematics - Algebraic GeometryLinear differential equationAlgebraic group0103 physical sciencesFOS: Mathematics010307 mathematical physics0101 mathematicsAlgebraic numberAlgebraic Geometry (math.AG)12H10 12H05 34M15 34M50 14L15MathematicsJournal of Pure and Applied Algebra
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Torsors for Difference Algebraic Groups

2016

We introduce a cohomology set for groups defined by algebraic difference equations and show that it classifies torsors under the group action. This allows us to compute all torsors for large classes of groups. We also develop some tools for difference algebraic geometry and present an application to the Galois theory of differential equations depending on a discrete parameter.

Pure mathematicsGroup (mathematics)Applied MathematicsGeneral Mathematics12H10 20G10 14L15 39A05Mathematics - Rings and AlgebrasCommutative Algebra (math.AC)Mathematics - Commutative AlgebraCohomologyAction (physics)Set (abstract data type)Mathematics - Algebraic GeometryRings and Algebras (math.RA)Mathematics::K-Theory and HomologyFOS: MathematicsAlgebraic numberAlgebraic Geometry (math.AG)Mathematics
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