Skip to main content


  • Lucia Di Vizio (a1), Charlotte Hardouin (a2) and Michael Wibmer (a3)

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.

Hide All
1. Arreche C. E., Computing the differential galois group of a one-parameter family of second order linear differential equations, 2012. arXiv:1208.2226.
2. Arreche C. E., A Galois-theoretic proof of the differential transcendence of the incomplete Gamma function, J. Algebra 389 (2013), 119127.
3. Beukers F. and Heckman G., Monodromy for the hypergeometric function n F n-1 , Invent. Math. 95(2) (1989), 325354.
4. Chatzidakis Z., Model theory of difference fields, in The Notre Dame Lectures, Lecture Notes Logic, Volume 18, pp. 4596 (Association for Symbolic Logic, Urbana, IL, 2005).
5. 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), 257311.
6. 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. 130137. (2012).
7. Cohn R. M., Difference Algebra (Interscience Publishers, John Wiley & Sons, New York, London, Sydeny, 1965).
8. Crew R., F-isocrystals and their monodromy groups, Annales Scientifiques de l’École Normale Supérieure. Quatrième Série 25(4) (1992), 429464.
9. 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).
10. 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.
11. Dreyfus T., Computing the galois group of some parameterized linear differential equation of order two, Proc. Amer. Math. Soc. 142(4) (2014), 11931207.
12. Dreyfus T., A density theorem for parameterized differential Galois theory, Pacific Journal of Mathematics 271(1) (2014), 87141.
13. Di Vizio L., Approche galoisienne de la transcendance différentielle, in Transcendance et irrationalité, SMF Journée Annuelle [SMF Annual Conference], pp. 120 (Société Mathématique de France, 2012).
14. Di Vizio L. and Hardouin C., Descent for differential Galois theory of difference equations: confluence and q-dependence, Pacific J. Math. 256(1) (2012), 79104.
15. Di Vizio L., Hardouin C. and Wibmer M., Difference Galois theory of linear differential equations, Adv. Math. 260 (2014), 158.
16. Dwork B., Generalized Hypergeometric Functions, Oxford Mathematical Monographs (The Clarendon Press, Oxford University Press, New York, 1990). Oxford Science Publications.
17. Dwork B., Gerotto G. and Sullivan F. J., An Introduction to G-Functions, Annals of Mathematics Studies, vol. 133 (Princeton University Press, 1994).
18. Gorchinskiy S. and Ovchinnikov A., Isomonodromic differential equations and differential categories, Journal de Mathématiques Pures et Appliquées 102 (2014), 4878, arXiv:1202.0927.
19. Hartmann J., On the inverse problem in differential Galois theory, J. Reine Angew. Math. 586 (2005), 2144.
20. Hardouin C. and Singer M. F., Differential Galois theory of linear difference equations, Math. Ann. 342(2) (2008), 333377.
21. Hrushovski E., The elementary theory of the Frobenius automorphisms, 2004. arXiv:math/0406514v1, updated version available from∼ehud/.
22. Humphreys J. E., Linear Algebraic Groups, Graduate Texts in Mathematics, vol. 21 (Springer-Verlag, New York, 1975).
23. Ishizaki K., Hypertranscendency of meromorphic solutions of a linear functional equation, Aequationes Math. 56(3) (1998), 271283.
24. Jantzen J. C., Representations of Algebraic Groups, Pure and Applied Mathematics, vol. 131 (Academic Press Inc., Boston, MA, 1987).
25. Katz N. M., Algebraic solutions of differential equations (p-curvature and the Hodge filtration), Invent. Math. 18 (1972), 1118.
26. Katz N. M., On the calculation of some differential Galois groups, Invent. Math. 87(1) (1987), 1361.
27. Kedlaya K. S., p-Adic Differential Equations, Cambridge Studies in Advanced Mathematics, vol. 125 (Cambridge University Press, Cambridge, 2010).
28. Kolchin E. R., Algebraic groups and algebraic dependence, Amer. J. Math. 90 (1968), 11511164.
29. Kolchin E. R., Differential Algebra and Algebraic Groups, Pure and Applied Mathematics, vol. 54 (Academic Press, New York, 1973).
30. Kovacic J. J., An algorithm for solving second order linear homogeneous differential equations, J. Symbolic Comput. 2(1) (1986), 343.
31. Kovacic J. J., An algorithm for solving second order linear homogeneous differential equations.∼ksda/PostedPapers/algorithm092305.pdf, 2005.
32. Kowalski P. and Pillay A., On algebraic 𝜎-groups, Trans. Amer. Math. Soc. 359(3) (2007), 13251337. (electronic).
33. Landesman P., Generalized differential Galois theory, Trans. Amer. Math. Soc. 360(8) (2008), 44414495.
34. Levin A., Difference Algebra, Algebra and Applications, vol. 8 (Springer, New York, 2008).
35. Milne J. S., Basic theory of affine group schemes, 2012. Available at
36. 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.
37. Minchenko A., Ovchinnikov A. and Singer M. F., Unipotent differential algebraic groups as parameterized differential Galois groups, J. Inst. Math. Jussieu 13(4) (2014), 671700.
38. Marker D. and Pillay A., Differential Galois theory. III. Some inverse problems, Illinois J. Math. 41(3) (1997), 453461.
39. Mitschi C. and Singer M. F., Monodromy groups of parameterized linear differential equations with regular singularities, Bull. Lond. Math. Soc. 44(5) (2012), 913930.
40. Ostrowski A., Sur les relations algébriques entre les intégrales indéfinies, Acta Math. 78 (1946), 315318.
41. Ovchinnikov A. and Wibmer M., inline-graphic $\unicode[STIX]{x1D70E}$ -Galois theory of linear difference equations (2013). International Mathematics Research Notices IMRN, 57 pages, doi:10.1093/imrn/rnu60, to appear. arXiv:1304.2649.
42. Pulita A., Frobenius structure for rank one p-adic differential equations, in Ultrametric Functional Analysis, Contemporary Mathematics, Volume 384, pp. 247258 (American Mathematical Society, Providence, RI, 2005).
43. 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).
44. van der Put M. and Singer M. F., Galois Theory of Linear Differential Equations (Springer-Verlag, Berlin, 2003).
45. Sanchez O. L., Relative inline-graphic $d$ -groups and differential Galois theory in several derivations, Trans. of AMS (to appear). arXiv:1212.0102.
46. 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.
47. Schémas en groupes (SGA 3). Tome II. Lecture notes in mathematics. Springer Verlag, 1970.
48. 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.
49. Sibuya Y., Linear Differential Equations in the Complex Domain: Problems of Analytic Continuation, Translations of Mathematical Monographs, Volume 82 (Providence, RI, 1990).
50. 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. 379420.
51. Singer M. F., Linear algebraic groups as parameterized Picard–Vessiot Galois groups, J. Algebra 373 (2013), 153161.
52. Springer T. A., Linear Algebraic Groups, 2nd ed., Modern Birkhäuser Classics (Birkhäuser Boston Inc., Boston, MA, 2009).
53. Stichtenoth H., Algebraic Function Fields and Codes, 2nd ed., Graduate Texts in Mathematics, vol. 254 (Springer-Verlag, Berlin, 2009).
54. Singer M. F. and Ulmer F., Galois groups of second and third order linear differential equations, J. Symbolic Comput. 16(1) (1993), 936.
55. Tarasov V. and Varchenko A., Difference equations compatible with trigonometric KZ differential equations. Internat. Math. Res. Notices (15) 801–829, 2000.
56. Waterhouse W. C., Introduction to Affine Group Schemes, Graduate Texts in Mathematics, vol. 66 (Springer-Verlag, New York, 1979).
57. Wibmer M., Affine difference algebraic groups. arXiv:1405.6603.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of the Institute of Mathematics of Jussieu
  • ISSN: 1474-7480
  • EISSN: 1475-3030
  • URL: /core/journals/journal-of-the-institute-of-mathematics-of-jussieu
Please enter your name
Please enter a valid email address
Who would you like to send this to? *



Full text views

Total number of HTML views: 0
Total number of PDF views: 38 *
Loading metrics...

Abstract views

Total abstract views: 204 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 17th January 2018. This data will be updated every 24 hours.