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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.

12H10 Difference algebra
58H05 Pseudogroups and differentiable groupoids
17B65 Infinite-dimensional Lie (super)algebras
Full Text: DOI EuDML
[1] Casale, G., Sur le groupoïde de Galois d’un feuilletage, (2004), Toulouse
[2] Casale, G., Enveloppe galoisienne d’une application rationnelle de \(\mathbb{P}^1,\) Publ. Mat., 50, 1, 191-202, (2006) · Zbl 1137.37022
[3] Franke, Charles H., Picard-Vessiot theory of linear homogeneous difference equations, Trans. Amer. Math. Soc., 108, 491-515, (1963) · Zbl 0116.02604
[4] Granier, A., Un \({D}\)-groupoïde de Galois pour les équations au \(q\)-différences, (2009), Toulouse
[5] Hardouin, Charlotte; Singer, Michael F., Differential Galois theory of linear difference equations, Math. Ann., 342, 2, 333-377, (2008) · Zbl 1163.12002
[6] Heiderlich, F., Infinitesimal Galois theory for \({D}\)-module fields
[7] Malgrange, B., Essays on geometry and related topics, Vol. 1, 2, 38, Le groupoïde de Galois d’un feuilletage, 465-501, (2001), Enseignement Math., Geneva · Zbl 1033.32020
[8] Morikawa, S.; Umemura, H, On a general Galois theory of difference equations II, Ann. Inst. Fourier, 59, 7, 2733-2771, (2009) · Zbl 1194.12006
[9] van der Put, Marius; Singer, Michael F., Galois theory of difference equations, 1666, (1997), Springer-Verlag, Berlin · Zbl 0930.12006
[10] Umemura, Hiroshi, Differential Galois theory of infinite dimension, Nagoya Math. J., 144, 59-135, (1996) · Zbl 0878.12002
[11] Umemura, Hiroshi, Galois theory of algebraic and differential equations, Nagoya Math. J., 144, 1-58, (1996) · Zbl 0885.12004
[12] Umemura, Hiroshi, Théories asymptotiques et équations de Painlevé, 14, Galois theory and Painlevé equations, 299-339, (2006), Soc. Math. France, Paris · Zbl 1156.34080
[13] Umemura, Hiroshi, Differential equations and quantum groups, 9, Invitation to Galois theory, 269-289, (2007), Eur. Math. Soc., Zürich · Zbl 1356.12006
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