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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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