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 , 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 , nous attachons son groupe de Galois qui est un groupe de transformations de coordonnées.
Accepted:
DOI: 10.5802/aif.2505
Classification: 12Hxx, 37Fxx, 58Hxx, 14Hxx
Keywords: General difference Galois theory, dynamical system, integrable dynamical system, Galois groupoid
@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}, zbl = {1194.12005}, mrnumber = {2649331}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2505/} }
TY - JOUR TI - On a general difference Galois theory I JO - Annales de l'Institut Fourier PY - 2009 DA - 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://zbmath.org/?q=an%3A1194.12005 UR - https://www.ams.org/mathscinet-getitem?mr=2649331 UR - https://doi.org/10.5802/aif.2505 DO - 10.5802/aif.2505 LA - en ID - AIF_2009__59_7_2709_0 ER -
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] Sur le groupoïde de Galois d’un feuilletage (2004) (Ph. D. Thesis)
[2] Enveloppe galoisienne d’une application rationnelle de , Publ. Mat., Tome 50 (2006) no. 1, pp. 191-202 | MR: 2325017 | Zbl: 1137.37022
[3] Picard-Vessiot theory of linear homogeneous difference equations, Trans. Amer. Math. Soc., Tome 108 (1963), pp. 491-515 | Article | MR: 155819 | Zbl: 0116.02604
[4] Un -groupoïde de Galois pour les équations au -différences (2009) (Ph. D. Thesis)
[5] Differential Galois theory of linear difference equations, Math. Ann., Tome 342 (2008) no. 2, pp. 333-377 | Article | MR: 2425146 | Zbl: 1163.12002
[6] Infinitesimal Galois theory for -module fields (in preparation)
[7] Le groupoïde de Galois d’un feuilletage, Essays on geometry and related topics, Vol. 1, 2 (Monogr. Enseign. Math.) Tome 38, Enseignement Math., Geneva, 2001, pp. 465-501 | MR: 1929336 | Zbl: 1033.32020
[8] On a general Galois theory of difference equations II, Ann. Inst. Fourier, Tome 59 (2009) no. 7, pp. 2733-2771 | Article | Numdam
[9] Galois theory of difference equations, Lecture Notes in Mathematics, Tome 1666, Springer-Verlag, Berlin, 1997 | MR: 1480919 | Zbl: 0930.12006
[10] Differential Galois theory of infinite dimension, Nagoya Math. J., Tome 144 (1996), pp. 59-135 http://projecteuclid.org/getRecord?id=euclid.nmj/1118771876 | MR: 1425592 | Zbl: 0878.12002
[11] Galois theory of algebraic and differential equations, Nagoya Math. J., Tome 144 (1996), pp. 1-58 http://projecteuclid.org/getRecord?id=euclid.nmj/1118771875 | MR: 1425591 | Zbl: 0885.12004
[12] Galois theory and Painlevé equations, Théories asymptotiques et équations de Painlevé (Sémin. Congr.) Tome 14, Soc. Math. France, Paris, 2006, pp. 299-339 | MR: 2353471 | Zbl: 1156.34080
[13] Invitation to Galois theory, Differential equations and quantum groups (IRMA Lect. Math. Theor. Phys.) Tome 9, Eur. Math. Soc., Zürich, 2007, pp. 269-289 | MR: 2322334
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