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


Zbl 1156.34080
Full Text: DOI Numdam EuDML


[1] Casale, G., Sur le groupoïde de Galois d’un feuilletage (2004)
[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 · doi:10.1090/S0002-9947-1963-0155819-3
[4] Granier, A., Un \({D}\)-groupoïde de Galois pour les équations au \(q\)-différences (2009)
[5] Hardouin, Charlotte; Singer, Michael F., Differential Galois theory of linear difference equations, Math. Ann., 342, 2, 333-377 (2008) · Zbl 1163.12002 · doi:10.1007/s00208-008-0238-z
[6] Heiderlich, F., Infinitesimal Galois theory for \({D}\)-module fields
[7] Malgrange, B., Essays on geometry and related topics, Vol. 1, 2, 38, 465-501 (2001) · 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 · doi:10.5802/aif.2506
[9] van der Put, Marius; Singer, Michael F., Galois theory of difference equations, 1666 (1997) · 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, 299-339 (2006) · Zbl 1156.34080
[13] Umemura, Hiroshi, Differential equations and quantum groups, 9, 269-289 (2007) · Zbl 1356.12006
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.