An Introduction to Difference Galois Theory

Part of the CRM Series in Mathematical Physics book series (CRM)


This article comes from notes written for my lectures at the summer school “Abecedarian of SIDE” held at the CRM (Montréal) in June 2016. They are intended to give a short introduction to difference Galois theory, leaving aside the technicalities.



This work has been partially supported by the LabEx PERSYVAL-Lab (ANR-11-LABX-0025-01) funded by the French program Investissement d’avenir.


  1. 1.
    B. Adamczewski, J.P. Bell, A problem about Mahler functions (2013), arXiv:1303.2019Google Scholar
  2. 2.
    J.P. Allouche, J. Shallit, Automatic Sequences: Theory, Applications, Generalizations (Cambridge University Press, Cambridge, 2003)CrossRefMATHGoogle Scholar
  3. 3.
    Y. André, Différentielles non commutatives et théorie de Galois différentielle ou aux différences. Ann. Sci. École Norm. Sup. (4) 34(5), 685–739 (2001)Google Scholar
  4. 4.
    C.E. Arreche, M.F. Singer, Galois groups for integrable and projectively integrable linear differential equations (2016), arXiv:1608.00015Google Scholar
  5. 5.
    P.G. Becker, k-regular power series and Mahler-type functional equations. J. Number Theory 49(3), 269–286 (1994)Google Scholar
  6. 6.
    D. Bertrand, Groupes algébriques et équations différentielles linéaires, in Séminaire Bourbaki. Astérisque, vol. 1991/1992, no. 206 (Société Mathématique de France, Paris, 1992), pp. 183–204Google Scholar
  7. 7.
    F. Beukers, Differential Galois theory, in From Number Theory to Physics, ed. by M. Waldschmidt, P. Moussa, J.M. Luck, C. Itzykson (Springer, Berlin/Heidelberg, 1992), pp. 413–439CrossRefGoogle Scholar
  8. 8.
    P.J. Cassidy, The classification of the semisimple differential algebraic groups and the linear semisimple differential algebraic Lie algebras. J. Algebra 121(1), 169–238 (1989)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    P.J. Cassidy, M.F. Singer, Galois theory of parameterized differential equations and linear differential algebraic groups, in Differential Equations and Quantum Groups, ed. by D. Bertrand, B. Enriquez, C. Mitschi, C. Sabbah, R. Schäfke. IRMA Lectures in Mathematics and Theoretical Physics, vol. 9 (European Mathematical Society, Zürich, 2007), pp. 113–155Google Scholar
  10. 10.
    L. Di Vizio, Approche galoisienne de la transcendance différentielle (2012), arXiv:1404.3611Google Scholar
  11. 11.
    T. Dreyfus, J. Roques, Galois groups of difference equations of order two on elliptic curves. SIGMA, Symmetry Integr. Geom. Methods Appl. 11, 003 (2015)Google Scholar
  12. 12.
    T. Dreyfus, C. Hardouin, J. Roques, Functional relations for solutions of q-difference equations (2016), arXiv:1603.06771Google Scholar
  13. 13.
    T. Dreyfus, C. Hardouin, J. Roques, Hypertranscendence of solutions of Mahler equations. J. Eur. Math. Soc. (to appear)Google Scholar
  14. 14.
    P. Dumas, Récurrences mahlériennes, suites automatiques, études asymptotiques. Ph.D. thesis, Université Bordeaux 1 (1993)Google Scholar
  15. 15.
    P.I. Etingof, Galois groups and connection matrices of q-difference equations. Electron. Res. Announc. Am. Math. Soc. 1(1), 1–9 (1995)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    R. Feng, On the computation of the Galois group of linear difference equations (2015), arXiv:1503.02239Google Scholar
  17. 17.
    C. Hardouin, M.F. Singer, Differential Galois theory of linear difference equations. Math. Ann. 342(2), 333–377 (2008)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    C. Hardouin, J. Sauloy, M.F. Singer, Galois Theories of Linear Difference Equations: An Introduction. Mathematical Surveys and Monographs, vol. 211 (American Mathematical Society, Providence, 2016)Google Scholar
  19. 19.
    P.A. Hendriks, An algorithm for computing a standard form for second-order linear q-difference equations. J. Pure Appl. Algebra 117/118, 331–352 (1997)Google Scholar
  20. 20.
    P.A. Hendriks, An algorithm determining the difference Galois group of second order linear difference equations. J. Symb. Comput. 26(4), 445–461 (1998)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    M. Mendès France, Nombres algébriques et théorie des automates. Enseign. Math. (2) 26(3-4), 193–199 (1980)Google Scholar
  22. 22.
    M. van der Put, Recent work on differential Galois theory, in Séminaire Bourbaki. Astérisque, vol. 1997/1998, no. 252 (Société Mathématique de France, Paris, 1998), pp. 341–367Google Scholar
  23. 23.
    M. van der Put, M.F. Singer, Galois Theory of Difference Equations. Lecture Notes in Mathematics, vol. 1666 (Springer, Berlin/Heidelberg, 1997)Google Scholar
  24. 24.
    M. van der Put, M.F. Singer, Galois Theory of Linear Differential Equations. Grundlehren der Mathematischen Wissenschaften, vol. 328 (Springer, Berlin/Heidelberg, 2003)Google Scholar
  25. 25.
    B. Randé, Équations fonctionnelles de Mahler et applications aux suites p-régulières. Ph.D. thesis, Université Bordeaux 1 (1992)Google Scholar
  26. 26.
    J. Roques, Galois groups of the basic hypergeometric equations. Pac. J. Math. 235(2), 303–322 (2008)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    J. Roques, Generalized basic hypergeometric equations. Invent. Math. 184(3), 499–528 (2011)ADSMathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    J. Roques, On classical irregular q-difference equations. Compos. Math. 148(5), 1624–1644 (2012)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    J. Roques, On the algebraic relations between Mahler functions. Trans. Am. Math. Soc. (to appear)Google Scholar
  30. 30.
    J. Sauloy, Systèmes aux q-différences singuliers réguliers: classification, matrice de connexion et monodromie. Ann. Inst. Fourier (Grenoble) 50(4), 1021–1071 (2000)Google Scholar
  31. 31.
    J. Sauloy, Galois theory of Fuchsian q-difference equations. Ann. Sci. École Norm. Sup. (4) 36(6), 925–968 (2003)Google Scholar
  32. 32.
    J. Sauloy, Introduction to Differential Galois Theory (2012). MATHGoogle Scholar
  33. 33.
    M.F. Singer, Galois theory of linear differential equations, in Algebraic Theory of Differential Equations, ed. by M.A.H. MacCallum, A.V. Mikhailov. London Mathematical Society Lecture Note Series, vol. 357 (Cambridge University Press, Cambridge, 2009), pp. 1–82Google Scholar

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© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Institut FourierUniversité Grenoble Alpes, UMR 5582 CNRS-UGAGièresFrance

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