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DIFFERENCE ALGEBRAIC RELATIONS AMONG SOLUTIONS OF LINEAR DIFFERENTIAL EQUATIONS

Part of: Differential equations in the complex domain Differential and difference algebra

Published online by Cambridge University Press:  17 April 2015

Lucia Di Vizio ,
Charlotte Hardouin  and
Michael Wibmer
Lucia Di Vizio
Affiliation:
Laboratoire de Mathématiques UMR8100, UVSQ, 45 avenue des États-Unis, 78035 Versailles cedex, France ([email protected])
Charlotte Hardouin
Affiliation:
Institut de Mathématiques de Toulouse, 118 route de Narbonne, 31062 Toulouse Cedex 9, France ([email protected])
Michael Wibmer
Affiliation:
Lehrstuhl für Mathematik (Algebra), RWTH Aachen, 52056 Aachen, Germany ([email protected])
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Abstract

We extend and apply the Galois theory of linear differential equations equipped with the action of an endomorphism. The Galois groups in this Galois theory are difference algebraic groups, and we use structure theorems for these groups to characterize the possible difference algebraic relations among solutions of linear differential equations. This yields tools to show that certain special functions are difference transcendent. One of our main results is a characterization of discrete integrability of linear differential equations with almost simple usual Galois group, based on a structure theorem for the Zariski dense difference algebraic subgroups of almost simple algebraic groups, which is a schematic version, in characteristic zero, of a result due to Z. Chatzidakis, E. Hrushovski, and Y. Peterzil.

Keywords

difference algebraGalois theoryalgebraic differential equationsdiscrete isomonodromy

MSC classification

Primary: 12H10: Difference algebra 12H20: Abstract differential equations 34M15: Algebraic aspects (differential-algebraic, hypertranscendence, group-theoretical) 33C99: None of the above, but in this section
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 16 , Issue 1 , February 2017 , pp. 59 - 119
Copyright
© Cambridge University Press 2015 

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References

Arreche, C. E., Computing the differential galois group of a one-parameter family of second order linear differential equations, 2012. arXiv:1208.2226.Google Scholar
Arreche, C. E., A Galois-theoretic proof of the differential transcendence of the incomplete Gamma function, J. Algebra 389 (2013), 119–127.Google Scholar
Beukers, F. and Heckman, G., Monodromy for the hypergeometric function n F n-1 , Invent. Math. 95(2) (1989), 325–354.Google Scholar
Chatzidakis, Z., Model theory of difference fields, in The Notre Dame Lectures, Lecture Notes Logic, Volume 18, pp. 45–96 (Association for Symbolic Logic, Urbana, IL, 2005).Google Scholar
Chatzidakis, Z., Hrushovski, E. and Peterzil, Y., Model theory of difference fields. II. Periodic ideals and the trichotomy in all characteristics, Proc. Lond. Math. Soc. (3) 85(2) (2002), 257–311.Google Scholar
Chen, S., Kauers, M. and Singer, M. F., Telescopers for rational and algebraic functions via residues, in Procedings of ISSAC 2012 (ed. van der Hoeven, J. and van Hoeij, M.), pp. 130–137. (2012).Google Scholar
Cohn, R. M., Difference Algebra (Interscience Publishers, John Wiley & Sons, New York, London, Sydeny, 1965).Google Scholar
Crew, R., F-isocrystals and their monodromy groups, Annales Scientifiques de l’École Normale Supérieure. Quatrième Série 25(4) (1992), 429–464.Google Scholar
Cassidy, P. J. and Singer, M. F., Galois theory of parameterized differential equations and linear differential algebraic groups, in Differential Equations and Quantum Groups, IRMA Lectures in Mathematics and Theoretical Physics, Volume 9, pp. 113–157 (Strasbourg, France, 2006).Google Scholar
Demazure, M., Le théorème d’existence, in Séminaire de Géométrie Algébrique du Bois-Marie (SGA), Tome 3, Structure des Schémas réductifs (Soc. Math. France, Paris, 1970). pp. Exp. No. XXV, 416–431.Google Scholar
Dreyfus, T., Computing the galois group of some parameterized linear differential equation of order two, Proc. Amer. Math. Soc. 142(4) (2014), 1193–1207.Google Scholar
Dreyfus, T., A density theorem for parameterized differential Galois theory, Pacific Journal of Mathematics 271(1) (2014), 87–141.Google Scholar
Di Vizio, L., Approche galoisienne de la transcendance différentielle, in Transcendance et irrationalité, SMF Journée Annuelle [SMF Annual Conference], pp. 1–20 (Société Mathématique de France, 2012).Google Scholar
Di Vizio, L. and Hardouin, C., Descent for differential Galois theory of difference equations: confluence and q-dependence, Pacific J. Math. 256(1) (2012), 79–104.CrossRefGoogle Scholar
Di Vizio, L., Hardouin, C. and Wibmer, M., Difference Galois theory of linear differential equations, Adv. Math. 260 (2014), 1–58.CrossRefGoogle Scholar
Dwork, B., Generalized Hypergeometric Functions, Oxford Mathematical Monographs (The Clarendon Press, Oxford University Press, New York, 1990). Oxford Science Publications.Google Scholar
Dwork, B., Gerotto, G. and Sullivan, F. J., An Introduction to G-Functions, Annals of Mathematics Studies, vol. 133 (Princeton University Press, 1994).Google Scholar
Gorchinskiy, S. and Ovchinnikov, A., Isomonodromic differential equations and differential categories, Journal de Mathématiques Pures et Appliquées 102 (2014), 48–78, arXiv:1202.0927.Google Scholar
Hartmann, J., On the inverse problem in differential Galois theory, J. Reine Angew. Math. 586 (2005), 21–44.Google Scholar
Hardouin, C. and Singer, M. F., Differential Galois theory of linear difference equations, Math. Ann. 342(2) (2008), 333–377.Google Scholar
Hrushovski, E., The elementary theory of the Frobenius automorphisms, 2004. arXiv:math/0406514v1, updated version available from http://www.ma.huji.ac.il/∼ehud/.Google Scholar
Humphreys, J. E., Linear Algebraic Groups, Graduate Texts in Mathematics, vol. 21 (Springer-Verlag, New York, 1975).Google Scholar
Ishizaki, K., Hypertranscendency of meromorphic solutions of a linear functional equation, Aequationes Math. 56(3) (1998), 271–283.Google Scholar
Jantzen, J. C., Representations of Algebraic Groups, Pure and Applied Mathematics, vol. 131 (Academic Press Inc., Boston, MA, 1987).Google Scholar
Katz, N. M., Algebraic solutions of differential equations (p-curvature and the Hodge filtration), Invent. Math. 18 (1972), 1–118.Google Scholar
Katz, N. M., On the calculation of some differential Galois groups, Invent. Math. 87(1) (1987), 13–61.Google Scholar
Kedlaya, K. S., p-Adic Differential Equations, Cambridge Studies in Advanced Mathematics, vol. 125 (Cambridge University Press, Cambridge, 2010).Google Scholar
Kolchin, E. R., Algebraic groups and algebraic dependence, Amer. J. Math. 90 (1968), 1151–1164.Google Scholar
Kolchin, E. R., Differential Algebra and Algebraic Groups, Pure and Applied Mathematics, vol. 54 (Academic Press, New York, 1973).Google Scholar
Kovacic, J. J., An algorithm for solving second order linear homogeneous differential equations, J. Symbolic Comput. 2(1) (1986), 3–43.Google Scholar
Kovacic, J. J., An algorithm for solving second order linear homogeneous differential equations. http://www.sci.ccny.cuny.edu/∼ksda/PostedPapers/algorithm092305.pdf, 2005.Google Scholar
Kowalski, P. and Pillay, A., On algebraic 𝜎-groups, Trans. Amer. Math. Soc. 359(3) (2007), 1325–1337. (electronic).Google Scholar
Landesman, P., Generalized differential Galois theory, Trans. Amer. Math. Soc. 360(8) (2008), 4441–4495.Google Scholar
Levin, A., Difference Algebra, Algebra and Applications, vol. 8 (Springer, New York, 2008).CrossRefGoogle Scholar
Milne, J. S., Basic theory of affine group schemes, 2012. Available at www.jmilne.org/math/.Google Scholar
Minchenko, A., Ovchinnikov, A. and Singer, M. F., Reductive linear differential algebraic groups and the galois groups of parameterized linear differential equations, Int. Math. Res. Not. IMRN (2014), Article ID rnt344, 61 pages. doi:10.1093/imrn/rnt344.Google Scholar
Minchenko, A., Ovchinnikov, A. and Singer, M. F., Unipotent differential algebraic groups as parameterized differential Galois groups, J. Inst. Math. Jussieu 13(4) (2014), 671–700.Google Scholar
Marker, D. and Pillay, A., Differential Galois theory. III. Some inverse problems, Illinois J. Math. 41(3) (1997), 453–461.Google Scholar
Mitschi, C. and Singer, M. F., Monodromy groups of parameterized linear differential equations with regular singularities, Bull. Lond. Math. Soc. 44(5) (2012), 913–930.CrossRefGoogle Scholar
Ostrowski, A., Sur les relations algébriques entre les intégrales indéfinies, Acta Math. 78 (1946), 315–318.Google Scholar
Ovchinnikov, A. and Wibmer, M., $\unicode[STIX]{x1D70E}$ -Galois theory of linear difference equations (2013). International Mathematics Research Notices IMRN, 57 pages, doi:10.1093/imrn/rnu60, to appear. arXiv:1304.2649.Google Scholar
Pulita, A., Frobenius structure for rank one p-adic differential equations, in Ultrametric Functional Analysis, Contemporary Mathematics, Volume 384, pp. 247–258 (American Mathematical Society, Providence, RI, 2005).Google Scholar
van der Put, M. and Singer, M. F., Galois Theory of Difference Equations, Lecture Notes in Mathematics, Volume 1666, viii+180 pages (Springer-Verlag, Berlin, 1997).Google Scholar
van der Put, M. and Singer, M. F., Galois Theory of Linear Differential Equations (Springer-Verlag, Berlin, 2003).CrossRefGoogle Scholar
Sanchez, O. L., Relative $d$ -groups and differential Galois theory in several derivations, Trans. of AMS (to appear). arXiv:1212.0102.Google Scholar
Schémas en groupes (SGA 3). Tome I. Propriétés générales des schémas en groupes. Documents Mathématiques (Paris) [Mathematical Documents (Paris)], 7. Société Mathématique de France, Paris, 2011. Séminaire de Géométrie Algébrique du Bois Marie 1962–64. Revised and annotated edition of the 1970 French original.Google Scholar
Schémas en groupes (SGA 3). Tome II. Lecture notes in mathematics. Springer Verlag, 1970.Google Scholar
Schémas en groupes (SGA 3). Tome III. Structure des schémas en groupes réductifs. Documents Mathématiques (Paris) [Mathematical Documents (Paris)], 8. Société Mathématique de France, Paris, 2011. Séminaire de Géométrie Algébrique du Bois Marie 1962–64. Revised and annotated edition of the 1970 French original.Google Scholar
Sibuya, Y., Linear Differential Equations in the Complex Domain: Problems of Analytic Continuation, Translations of Mathematical Monographs, Volume 82 (Providence, RI, 1990).Google Scholar
Singer, M. F., Algebraic solutions of nth order linear differential equations, in Proceedings of the Queen’s Number Theory Conference, 1979 (Kingston, Ont., 1979, Queen’s Papers in Pure and Appl. Math., Volume 54, pp. 379–420.Google Scholar
Singer, M. F., Linear algebraic groups as parameterized Picard–Vessiot Galois groups, J. Algebra 373 (2013), 153–161.CrossRefGoogle Scholar
Springer, T. A., Linear Algebraic Groups, 2nd ed., Modern Birkhäuser Classics (Birkhäuser Boston Inc., Boston, MA, 2009).Google Scholar
Stichtenoth, H., Algebraic Function Fields and Codes, 2nd ed., Graduate Texts in Mathematics, vol. 254 (Springer-Verlag, Berlin, 2009).Google Scholar
Singer, M. F. and Ulmer, F., Galois groups of second and third order linear differential equations, J. Symbolic Comput. 16(1) (1993), 9–36.Google Scholar
Tarasov, V. and Varchenko, A., Difference equations compatible with trigonometric KZ differential equations. Internat. Math. Res. Notices (15) 801–829, 2000.Google Scholar
Waterhouse, W. C., Introduction to Affine Group Schemes, Graduate Texts in Mathematics, vol. 66 (Springer-Verlag, New York, 1979).Google Scholar
Wibmer, M., Affine difference algebraic groups. arXiv:1405.6603.Google Scholar