Table of contents for this issue | Previous article | Next article
Shuji Morikawa; Hiroshi Umemura
On a general difference Galois theory II
(Théorie de Galois générale aux différences II)
Annales de l'institut Fourier, 59 no. 7 (2009), p. 2733-2771, doi: 10.5802/aif.2506
Article PDF | Reviews MR 2649332 | Zbl 1194.12006 | 1 citation in Cedram
Class. Math.: 12Hxx, 37Fxx, 58Hxx, 14Hxx
Keywords: General difference Galois theory, dynamical system, integrable dynamical system, Galois groupoid
[1] G. Casale, Sur le groupoïde de Galois d’un feuilletage, Ph. D. Thesis, Université Paul Sabatier, Toulouse, http://doctorants.picard.ups-tlse.fr/theses.htm, 2004 Article
[2] G. Casale, “Enveloppe galoisienne d’une application rationnelle de $\mathbb{P}^1$”, Publ. Mat. 50 (2006), p. 191-202 MR 2325017 | Zbl 1137.37022
[3] M. Demazure, “Sous-groupes algébriques de rang maximum du groupe de Cremona”, Ann. Sci. École Norm. Sup. (4) 3 (1970), p. 507-588 Numdam | MR 284446 | Zbl 0223.14009
[4] A. Grothendieck, Techniques de constructions et théorème d’existence en géométrie algébrique III, Préschémas quotients, Séminaire Bourbaki, Vol. 6, Soc. Math. France, 1995, p. Exp. No. 212, 99–118 Numdam | Zbl 0235.14007
[5] E. Kolchin, Differential algebra and algebraic groups, Pure and Applied Mathematics, Academic Press, New York-London, 1973 MR 568864 | Zbl 0264.12102
[6] S. Lie, Théorie des Transformationsgruppen, Teubner, 1930 JFM 23.0364.01
[7] B. Malgrange, Le groupoïde de Galois d’un feuilletage, Essays on geometry and related topics, Vol. 1, 2, Monogr. Enseign. Math. 38, Enseignement Math., 2001, p. 465–501 MR 1929336 | Zbl 1033.32020
[8] S. Morikawa, “On a general Galois theory of difference equations I”, Ann. Inst. Fourier, Grenoble 2010, à paraître
[9] J. H. Silverman, The arithmetic of dynamical systems, Graduate Texts in Mathematics 241, Springer, New York, 2007 MR 2316407 | Zbl 1130.37001
[10] H. Umemura, “Differential Galois theory of infinite dimension”, Nagoya Math. J. 144 (1996), p. 59-135 Article | MR 1425592 | Zbl 0878.12002
[11] H. Umemura, Galois theory and Painlevé equations, Théories asymptotiques et équations de Painlevé, Sémin. Congr. 14, Soc. Math. France, 2006, p. 299–339 MR 2353471 | Zbl 1156.34080
[12] H. Umemura, Invitation to Galois theory, Differential equations and quantum groups, IRMA Lect. Math. Theor. Phys. 9, Eur. Math. Soc., Zürich, 2007, p. 269–289 MR 2322334
[13] H. Umemura, “Sur l’équivalence des théories de Galois différentielles générales”, C. R. Math. Acad. Sci. Paris 346 (2008) no. 21-22, p. 1155-1158 Article | MR 2464256 | Zbl pre05374913
[14] H. Umemura, On the definition of Galois groupoid, Differential Equations and Singularities, 60 years of J.M. Aroca, Soc. Math. France, 2010
[15] A. P. Veselov, “Integrable mappings”, Russian Math. Surveys 46 (1991), p. 1-51 Article | MR 1160332 | Zbl 0785.58027
Shuji Morikawa; Hiroshi Umemura
On a general difference Galois theory II
(Théorie de Galois générale aux différences II)
Annales de l'institut Fourier, 59 no. 7 (2009), p. 2733-2771, doi: 10.5802/aif.2506
Article PDF | Reviews MR 2649332 | Zbl 1194.12006 | 1 citation in Cedram
Class. Math.: 12Hxx, 37Fxx, 58Hxx, 14Hxx
Keywords: General difference Galois theory, dynamical system, integrable dynamical system, Galois groupoid
Résumé - Abstract
We apply the General Galois Theory of difference equations introduced in the first part to concrete examples. The General Galois Theory allows us to define a discrete dynamical system being infinitesimally solvable, which is a finer notion than being integrable. We determine all the infinitesimally solvable discrete dynamical systems on the compact Riemann surfaces.
Bibliography
[2] G. Casale, “Enveloppe galoisienne d’une application rationnelle de $\mathbb{P}^1$”, Publ. Mat. 50 (2006), p. 191-202 MR 2325017 | Zbl 1137.37022
[3] M. Demazure, “Sous-groupes algébriques de rang maximum du groupe de Cremona”, Ann. Sci. École Norm. Sup. (4) 3 (1970), p. 507-588 Numdam | MR 284446 | Zbl 0223.14009
[4] A. Grothendieck, Techniques de constructions et théorème d’existence en géométrie algébrique III, Préschémas quotients, Séminaire Bourbaki, Vol. 6, Soc. Math. France, 1995, p. Exp. No. 212, 99–118 Numdam | Zbl 0235.14007
[5] E. Kolchin, Differential algebra and algebraic groups, Pure and Applied Mathematics, Academic Press, New York-London, 1973 MR 568864 | Zbl 0264.12102
[6] S. Lie, Théorie des Transformationsgruppen, Teubner, 1930 JFM 23.0364.01
[7] B. Malgrange, Le groupoïde de Galois d’un feuilletage, Essays on geometry and related topics, Vol. 1, 2, Monogr. Enseign. Math. 38, Enseignement Math., 2001, p. 465–501 MR 1929336 | Zbl 1033.32020
[8] S. Morikawa, “On a general Galois theory of difference equations I”, Ann. Inst. Fourier, Grenoble 2010, à paraître
[9] J. H. Silverman, The arithmetic of dynamical systems, Graduate Texts in Mathematics 241, Springer, New York, 2007 MR 2316407 | Zbl 1130.37001
[10] H. Umemura, “Differential Galois theory of infinite dimension”, Nagoya Math. J. 144 (1996), p. 59-135 Article | MR 1425592 | Zbl 0878.12002
[11] H. Umemura, Galois theory and Painlevé equations, Théories asymptotiques et équations de Painlevé, Sémin. Congr. 14, Soc. Math. France, 2006, p. 299–339 MR 2353471 | Zbl 1156.34080
[12] H. Umemura, Invitation to Galois theory, Differential equations and quantum groups, IRMA Lect. Math. Theor. Phys. 9, Eur. Math. Soc., Zürich, 2007, p. 269–289 MR 2322334
[13] H. Umemura, “Sur l’équivalence des théories de Galois différentielles générales”, C. R. Math. Acad. Sci. Paris 346 (2008) no. 21-22, p. 1155-1158 Article | MR 2464256 | Zbl pre05374913
[14] H. Umemura, On the definition of Galois groupoid, Differential Equations and Singularities, 60 years of J.M. Aroca, Soc. Math. France, 2010
[15] A. P. Veselov, “Integrable mappings”, Russian Math. Surveys 46 (1991), p. 1-51 Article | MR 1160332 | Zbl 0785.58027
© Annales de L'Institut Fourier - ISSN (électronique) : 1777-5310