Galois theories of $q$-difference equations: comparison theorems
Confluentes Mathematici, Volume 12 (2020) no. 2, pp. 11-35.

We establish some comparison results among the different parameterized Galois theories for $q$-difference equations, completing the work [4], that addresses the problem in the case without parameters. Our main result is the link between the abstract parameterized Galois theories, that give information on the differential properties of abstract solutions of $q$-difference equations, and the properties of meromorphic solutions of such equations. Notice that a linear $q$-difference equation with meromorphic coefficients always admits a basis of meromorphic solutions, as proven in [25].

Revised:
Accepted:
Published online:
DOI: 10.5802/cml.66
Classification: 39A13,  12H10
Keywords: Galois group; $q$-difference equations; differential Tannakian categories; Kolchin differential groups.
Di Vizio, Lucia 1; Hardouin, Charlotte 2

1 Université Paris-Saclay, UVSQ, CNRS, Laboratoire de mathématiques de Versailles, 78000, Versailles, France
2 Institut de Mathématiques de Toulouse, 118 route de Narbonne, 31062 Toulouse Cedex 9, France
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Di Vizio, Lucia; Hardouin, Charlotte. Galois theories of $q$-difference equations: comparison theorems. Confluentes Mathematici, Volume 12 (2020) no. 2, pp. 11-35. doi : 10.5802/cml.66. http://www.numdam.org/articles/10.5802/cml.66/

[1] André, Y. Différentielles non commutatives et théorie de Galois différentielle ou aux différences, Annales Scientifiques de l’École Normale Supérieure. Quatrième Série, Volume 34 (2001) no. 5, pp. 685-739 | DOI | Numdam | Zbl

[2] Barkatou, M.; Cluzeau, T.; Di Vizio, L.; Weil, J.-A. Reduced forms of linear differential systems and the intrinsic Galois-Lie algebra of Katz, 2019 | arXiv

[3] Cassidy, P.J.; Singer, M. F. Galois Theory of Parameterized Differential Equations and Linear Differential Algebraic Groups, Differential Equations and Quantum Groups (IRMA Lectures in Mathematics and Theoretical Physics), Volume 9, 2006, pp. 113-157 | DOI

[4] Chatzidakis, Z.; Hardouin, C.; Singer, M. F. On the Definitions of Difference Galois Groups, Model Theory with applications to algebra and analysis, I and II, Cambridge University Press, 2008, pp. 73-109 | Zbl

[5] Cohn, R. M. Difference algebra, Interscience Publishers John Wiley & Sons, New York-London-Sydney, 1965, xiv+355 pages | Zbl

[6] Deligne, P. Catégories tannakiennes, in The Grothendieck Festschrift, Vol II (Prog.Math.), Volume 87, Birkhaäuser, Boston, 1990, pp. 111-195 | Zbl

[7] Deligne, P.; Milne, J. S.; Ogus, A.; Shih, K. Hodge cycles, motives, and Shimura varieties, Lecture Notes in Mathematics, 900, Springer-Verlag, Berlin, 1982 | MR | Zbl

[8] Di Vizio, L. Arithmetic theory of $q$-difference equations. The $q$-analogue of Grothendieck-Katz’s conjecture on $p$-curvatures, Inventiones Mathematicae, Volume 150 (2002) no. 3, pp. 517-578 | DOI | MR | Zbl

[9] Di Vizio, L.; Hardouin, C. Intrinsic approach to Galois theory of $q$-difference equations, Memoirs of the AMS (2019) (with a preface to Part IV by Anne Granier. To appear)

[10] Di Vizio, L.; Hardouin, C. Descent for differential Galois theory of difference equations. Confluence and $q$-dependency, Pacific Journal of Mathematics (2012) no. 1, pp. 79-104 | DOI | Zbl

[11] Di Vizio, L.; Ramis, J.-P.; Sauloy, J.; Zhang, C. Équations aux $q$-différences, Gazette des Mathématiciens, Volume 96 (2003), pp. 20-49 | Zbl

[12] Dreyfus, T. Building meromorphic solutions of $q$-difference equations using a Borel-Laplace summation, Int. Math. Res. Not. IMRN (2015) no. 15, pp. 6562-6587 | DOI | MR | Zbl

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

[14] Gillet, H. Differential algebra—a scheme theory approach, Differential algebra and related topics (Newark, NJ, 2000), World Sci. Publ., River Edge, NJ, 2002, pp. 95-123 | DOI | MR | Zbl

[15] Gillet, H.; Gorchinskiy, S.; Ovchinnikov, A. Parameterized Picard-Vessiot extensions and Atiyah extensions, Adv. Math., Volume 238 (2013), pp. 322-411 | DOI | MR | Zbl

[16] Hardouin, C.; Singer, M. F. Differential Galois theory of linear difference equations, Mathematische Annalen, Volume 342 (2008) no. 2, pp. 333-377 | DOI | MR | Zbl

[17] Kamensky, M. Model theory and the Tannakian formalism, 2010

[18] Kaplansky, I. An introduction to differential algebra, Hermann, Paris, 1957 | Zbl

[19] Katz, N. M. On the calculation of some differential Galois groups, Invent. Math., Volume 87 (1987) no. 1, pp. 13-61 | DOI | MR | Zbl

[20] Kolchin, E.R. Differential algebra and algebraic groups, Pure and applied mathematics, 54, Academic Press, New York and London, 1973 | MR

[21] Kovacic, J. J. Differential schemes, Differential algebra and related topics (Newark, NJ, 2000), World Sci. Publ., River Edge, NJ, 2002, pp. 71-94 | DOI | Zbl

[22] Levin, A. Difference algebra, Algebra and Applications, 8, Springer, New York, 2008 | MR | Zbl

[23] Ovchinnikov, A. Differential Tannakian categories, Journal of Algebra, Volume 321 (2009) no. 10, pp. 3043-3062 | DOI | MR | Zbl

[24] Peón Nieto, A. On $\sigma \delta$-Picard-Vessiot extensions, Comm. Algebra, Volume 39 (2011) no. 4, pp. 1242-1249 | MR | Zbl

[25] Praagman, C. Fundamental solutions for meromorphic linear difference equations in the complex plane, and related problems, Journal für die Reine und Angewandte Mathematik, Volume 369 (1986), pp. 101-109 | MR | Zbl

[26] Sauloy, J. Systèmes aux $q$-différences singuliers réguliers: classification, matrice de connexion et monodromie, Annales de l’Institut Fourier, Volume 50 (2000) no. 4, pp. 1021-1071 | DOI | Numdam | MR | Zbl

[27] Sauloy, J. Galois theory of Fuchsian $q$-difference equations, Ann. Sci. École Norm. Sup. (4), Volume 36 (2003) no. 6, pp. 925-968 | DOI | Numdam | MR | Zbl

[28] Sauloy, J. Galois theory of Fuchsian $q$-difference equations, Annales Scientifiques de l’École Normale Supérieure. Quatrième Série, Volume 36 (2004) no. 6, pp. 925-968

[29] van der Put, M.; Singer, M. F. Galois theory of difference equations, Springer-Verlag, Berlin, 1997, viii+180 pages | DOI | Zbl

[30] van der Put, M.; Singer, M. F. Galois theory of linear differential equations, Springer-Verlag, Berlin, 2003, viii+180 pages | DOI | Zbl

[31] Waterhouse, W. C. Introduction to affine group schemes, Graduate Texts in Mathematics, 66, Springer-Verlag, New York, 1979 | MR | Zbl

[32] Wibmer, M. A Chevalley theorem for difference equations, Math. Ann., Volume 354 (2012) no. 4, pp. 1369-1396 | DOI | MR | Zbl

[33] Wibmer, M. Existence of $\partial$-parameterized Picard-Vessiot extensions over fields with algebraically closed constants, J. Algebra, Volume 361 (2012), pp. 163-171 | DOI | MR | Zbl

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