no. 7
On a general difference Galois theory I
Morikawa, Shuji1
1 Nagoya University Graduate School of Mathematics Nagoya (Japan)
Annales de l'Institut Fourier, Volume 59 (2009) no. 7, pp. 2709-2732.

We know well difference Picard-Vessiot theory, Galois theory of linear difference equations. We propose a general Galois theory of difference equations that generalizes Picard-Vessiot theory. For every difference field extension of characteristic 0, we attach its Galois group, which is a group of coordinate transformation.

La théorie de Picard-Vessiot aux différences, la théorie de Galois des équations aux différences linéaires, est bien connue. Nous proposons une théorie de Galois des équations aux différences générales qui généralise la théorie de Picard-Vessiot. Pour toute extension de corps aux différences de caractéristique 0, nous attachons son groupe de Galois qui est un groupe de transformations de coordonnées.

DOI: 10.5802/aif.2505
Classification: 12Hxx, 37Fxx, 58Hxx, 14Hxx
Keywords: General difference Galois theory, dynamical system, integrable dynamical system, Galois groupoid
Morikawa, Shuji 1

1 Nagoya University Graduate School of Mathematics Nagoya (Japan) 
@article{AIF_2009__59_7_2709_0,
     author = {Morikawa, Shuji},
     title = {On a general difference {Galois} theory {I}},
     journal = {Annales de l'Institut Fourier},
     pages = {2709--2732},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {59},
     number = {7},
     year = {2009},
     doi = {10.5802/aif.2505},
     mrnumber = {2649331},
     zbl = {1194.12005},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2505/}
}
TY  - JOUR
AU  - Morikawa, Shuji
TI  - On a general difference Galois theory I
JO  - Annales de l'Institut Fourier
PY  - 2009
SP  - 2709
EP  - 2732
VL  - 59
IS  - 7
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.2505/
UR  - https://www.ams.org/mathscinet-getitem?mr=2649331
UR  - https://zbmath.org/?q=an%3A1194.12005
UR  - https://doi.org/10.5802/aif.2505
DO  - 10.5802/aif.2505
LA  - en
ID  - AIF_2009__59_7_2709_0
ER  - 
%0 Journal Article
%A Morikawa, Shuji
%T On a general difference Galois theory I
%J Annales de l'Institut Fourier
%D 2009
%P 2709-2732
%V 59
%N 7
%I Association des Annales de l’institut Fourier
%U https://doi.org/10.5802/aif.2505
%R 10.5802/aif.2505
%G en
%F AIF_2009__59_7_2709_0
Morikawa, Shuji. On a general difference Galois theory I. Annales de l'Institut Fourier, Volume 59 (2009) no. 7, pp. 2709-2732. doi : 10.5802/aif.2505. https://aif.centre-mersenne.org/articles/10.5802/aif.2505/

[1] Casale, G. Sur le groupoïde de Galois d’un feuilletage, Université Paul Sabatier (2004) (Ph. D. Thesis)

[2] Casale, G. Enveloppe galoisienne d’une application rationnelle de ℙ 1 , Publ. Mat., Volume 50 (2006) no. 1, pp. 191-202 | MR | Zbl

[3] Franke, Charles H. Picard-Vessiot theory of linear homogeneous difference equations, Trans. Amer. Math. Soc., Volume 108 (1963), pp. 491-515 | DOI | MR | Zbl

[4] Granier, A. Un D -groupoïde de Galois pour les équations au q -différences, Université Paul Sabatier (2009) (Ph. D. Thesis)

[5] Hardouin, Charlotte; Singer, Michael F. Differential Galois theory of linear difference equations, Math. Ann., Volume 342 (2008) no. 2, pp. 333-377 | DOI | MR | Zbl

[6] Heiderlich, F. Infinitesimal Galois theory for D -module fields (in preparation)

[7] Malgrange, B. Le groupoïde de Galois d’un feuilletage, Essays on geometry and related topics, Vol. 1, 2 (Monogr. Enseign. Math.), Volume 38, Enseignement Math., Geneva, 2001, pp. 465-501 | MR | Zbl

[8] Morikawa, S.; Umemura, H On a general Galois theory of difference equations II, Ann. Inst. Fourier, Volume 59 (2009) no. 7, pp. 2733-2771 | DOI | Numdam

[9] van der Put, Marius; Singer, Michael F. Galois theory of difference equations, Lecture Notes in Mathematics, 1666, Springer-Verlag, Berlin, 1997 | MR | Zbl

[10] Umemura, Hiroshi Differential Galois theory of infinite dimension, Nagoya Math. J., Volume 144 (1996), pp. 59-135 http://projecteuclid.org/getRecord?id=euclid.nmj/1118771876 | MR | Zbl

[11] Umemura, Hiroshi Galois theory of algebraic and differential equations, Nagoya Math. J., Volume 144 (1996), pp. 1-58 http://projecteuclid.org/getRecord?id=euclid.nmj/1118771875 | MR | Zbl

[12] Umemura, Hiroshi Galois theory and Painlevé equations, Théories asymptotiques et équations de Painlevé (Sémin. Congr.), Volume 14, Soc. Math. France, Paris, 2006, pp. 299-339 | MR | Zbl

[13] Umemura, Hiroshi Invitation to Galois theory, Differential equations and quantum groups (IRMA Lect. Math. Theor. Phys.), Volume 9, Eur. Math. Soc., Zürich, 2007, pp. 269-289 | MR

Cited by Sources: