Skip to main content
SpringerLink
Log in

An Introduction to Difference Galois Theory

  • Chapter
  • First Online:
Symmetries and Integrability of Difference Equations

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

  • 707 Accesses

Abstract

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 is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    This means that, for any s ∈ S, the growth of the entries of \(\mathfrak{y}(z)\) as z tends to s along any sector of finite aperture and centered at s is at most polynomial.

References

  1. B. Adamczewski, J.P. Bell, A problem about Mahler functions (2013), arXiv:1303.2019

    Google Scholar 

  2. J.P. Allouche, J. Shallit, Automatic Sequences: Theory, Applications, Generalizations (Cambridge University Press, Cambridge, 2003)

    Book  MATH  Google Scholar 

  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. C.E. Arreche, M.F. Singer, Galois groups for integrable and projectively integrable linear differential equations (2016), arXiv:1608.00015

    Google Scholar 

  5. P.G. Becker, k-regular power series and Mahler-type functional equations. J. Number Theory 49(3), 269–286 (1994)

    Google Scholar 

  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. 183–204

    Google Scholar 

  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. 413–439

    Chapter  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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. 113–155

    Google Scholar 

  10. L. Di Vizio, Approche galoisienne de la transcendance différentielle (2012), arXiv:1404.3611

    Google Scholar 

  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. T. Dreyfus, C. Hardouin, J. Roques, Functional relations for solutions of q-difference equations (2016), arXiv:1603.06771

    Google Scholar 

  13. T. Dreyfus, C. Hardouin, J. Roques, Hypertranscendence of solutions of Mahler equations. J. Eur. Math. Soc. (to appear)

    Google Scholar 

  14. P. Dumas, Récurrences mahlériennes, suites automatiques, études asymptotiques. Ph.D. thesis, Université Bordeaux 1 (1993)

    Google Scholar 

  15. P.I. Etingof, Galois groups and connection matrices of q-difference equations. Electron. Res. Announc. Am. Math. Soc. 1(1), 1–9 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  16. R. Feng, On the computation of the Galois group of linear difference equations (2015), arXiv:1503.02239

    Google Scholar 

  17. C. Hardouin, M.F. Singer, Differential Galois theory of linear difference equations. Math. Ann. 342(2), 333–377 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  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. 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. P.A. Hendriks, An algorithm determining the difference Galois group of second order linear difference equations. J. Symb. Comput. 26(4), 445–461 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  21. M. Mendès France, Nombres algébriques et théorie des automates. Enseign. Math. (2) 26(3-4), 193–199 (1980)

    Google Scholar 

  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. 341–367

    Google Scholar 

  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. 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. B. Randé, Équations fonctionnelles de Mahler et applications aux suites p-régulières. Ph.D. thesis, Université Bordeaux 1 (1992)

    Google Scholar 

  26. J. Roques, Galois groups of the basic hypergeometric equations. Pac. J. Math. 235(2), 303–322 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  27. J. Roques, Generalized basic hypergeometric equations. Invent. Math. 184(3), 499–528 (2011)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  28. J. Roques, On classical irregular q-difference equations. Compos. Math. 148(5), 1624–1644 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  29. J. Roques, On the algebraic relations between Mahler functions. Trans. Am. Math. Soc. (to appear)

    Google Scholar 

  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. J. Sauloy, Galois theory of Fuchsian q-difference equations. Ann. Sci. École Norm. Sup. (4) 36(6), 925–968 (2003)

    Google Scholar 

  32. J. Sauloy, Introduction to Differential Galois Theory (2012). https://www.math.univ-toulouse.fr/~sauloy/

    MATH  Google Scholar 

  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. 1–82

    Google Scholar 

Download references

Acknowledgements

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

Author information

Authors and Affiliations

  1. Institut Fourier, Université Grenoble Alpes, UMR 5582 CNRS-UGA, 100, rue des maths, 38610, Gières, France

    Julien Roques

Authors
  1. Julien Roques
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Julien Roques .

Editor information

Editors and Affiliations

  1. Mathematics and Physics Departments, and INFN, Sezione Roma Tre, Roma Tre University, Roma, Italy

    Decio Levi

  2. Collège Ahuntsic, Montréal, Québec, Canada

    Raphaël Rebelo

  3. Centre de Recherches Mathématiques and Département de Mathématiques et de Statistique, Université de Montréal, Montréal, Québec, Canada

    Pavel Winternitz

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this chapter

Cite this chapter

Roques, J. (2017). An Introduction to Difference Galois Theory. In: Levi, D., Rebelo, R., Winternitz, P. (eds) Symmetries and Integrability of Difference Equations. CRM Series in Mathematical Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-56666-5_8

Download citation

Publish with us

Policies and ethics

Search

Navigation