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Zbl 1194.12005
Morikawa, Shuji
On a general difference Galois theory. I. (English)
[J] Ann. Inst. Fourier 59, No. 7, 2709-2732 (2009). ISSN 0373-0956; ISSN 1777-5310/e

The author has developed a new Galois theory of difference equations, being based on {\it 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 \par Conjecture: The canonical morphism from $\text{Lie}(\text{Inf-gal}(L/k))$ to $\text{Lie}(\text{Inf-gal}(M/k))$ is surjective.
[Nikolay Vasilye Grigorenko (Ky{\"\i}v)]

MSC 2000:: *12H10 Difference algebra
58H05 Pseudogroups on manifolds
17B65 Infinite-dimensional Lie algebras

Keywords: difference equations; Galois theory; Galois groupoid; infinite-dimensional Lie algebras

Citations: Zbl 1156.34080

Cited in: Zbl 1252.12007 Zbl 1194.12006

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