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On a general difference Galois theory. I. (English) Zbl 1194.12005
The author has developed a new Galois theory of difference equations, being based on H. Umemura’s ideas [“Galois theory and Painlevé equations”, Théories asymptotiques et équations de Painlevé, Soc. Math. France. Sémin. Congr. 14, 299–339 (2006; Zbl 1156.34080)]. This theory generalizes Picard-Vessiot and Galois theory of linear difference equations. It attaches to an arbitrary difference field extension \(L/k\) of characteristic 0 a formal group \(\text{Inf-gal}(L/k)\) of infinite dimension in general and of particular type called a Lie-Ritt functor, which is a group functor of coordinate transformations defined by a system of partial differential equations. Unfortunately, in this theory Galois correspondence is not expected. As replacement to it for a tower of difference field extensions \(L/M/k\) serves the following
Conjecture: The canonical morphism from \(\text{Lie}(\text{Inf-gal}(L/k))\) to \(\text{Lie}(\text{Inf-gal}(M/k))\) is surjective.

MSC:
12H10 Difference algebra
58H05 Pseudogroups and differentiable groupoids
17B65 Infinite-dimensional Lie (super)algebras
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References:
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