Advertisement

Inventiones mathematicae

, Volume 213, Issue 1, pp 139–203 | Cite as

On the nature of the generating series of walks in the quarter plane

  • Thomas Dreyfus
  • Charlotte Hardouin
  • Julien RoquesEmail author
  • Michael F. Singer
Article

Abstract

In the present paper, we introduce a new approach, relying on the Galois theory of difference equations, to study the nature of the generating series of walks in the quarter plane. Using this approach, we are not only able to recover many of the recent results about these series, but also to go beyond them. For instance, we give for the first time hypertranscendency results, i.e., we prove that certain of these generating series do not satisfy any nontrivial nonlinear algebraic differential equation with rational function coefficients.

Mathematics Subject Classification

05A15 30D05 39A06 

References

  1. 1.
    Bernardi, O., Bousquet-Mélou, M., Raschel, K.: Counting quadrant walks via Tutte’s invariant method (extended abstract), to appear in Proceedings of FPSAC 2015, Discrete Mathematics and Theoretical Computer Science (2016)Google Scholar
  2. 2.
    Bousquet-Mélou, M., Mishna, M.: Walks with small steps in the quarter plane. In: Algorithmic Probability and Combinatorics, Contemporary Mathematics, vol. 520, pp. 1–39. American Mathematical Society, Providence (2010)Google Scholar
  3. 3.
    Bostan, A., Raschel, K., Salvy, B.: Non-D-finite excursions in the quarter plane. J. Comb. Theory Ser. A 121, 45–63 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bostan, A., van Hoeij, M., Kauers, M.: The complete generating function for Gessel walks is algebraic. Proc. Am. Math. Soc. 138(9), 3063–3078 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cox, D.A., Little, J., O’Shea, D.: Using algebraic geometry. In: Graduate Texts in Mathematics, vol. 185, 2nd edn. Springer, New York (2005)Google Scholar
  6. 6.
    Chen, S., Singer, M.F.: Residues and telescopers for bivariate rational functions. Adv. Appl. Math. 49(2), 111–133 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Dreyfus, T., Hardouin, C., Roques, J.: Hypertranscendance of solutions of Mahler equations. To appear in J. Eur. Math. SocGoogle Scholar
  8. 8.
    Duistermaat, J.: Discrete Integrable Systems: Qrt Maps and Elliptic Surfaces, Springer Monographs in Mathematics, vol. 304. Springer, New York (2010)zbMATHGoogle Scholar
  9. 9.
    Fayolle, G., Iasnogorodski, R., Malyshev, V.: Random walks in the quarter-plane: algebraic methods, boundary value problems and applications. In: Applications of Mathematics, vol. 40. Springer, New York (1999)Google Scholar
  10. 10.
    Fayolle, G., Raschel, K.: On the holonomy or algebraicity of generating functions counting lattice walks in the quarter-plane. Markov Process. Relat. Fields 16(3), 485–496 (2010)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Hardouin, C.: Hypertranscendance des systèmes aux différences diagonaux. Compos. Math. 144(3), 565–581 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Hardouin, C.: Galoisian approach to differential transcendence. In: Galois Theories of Linear Difference Equations: An Introduction, Mathematical Surveys and Monographs, vol. 211, pp. 43–102. American Mathematical Society, Providence (2016)Google Scholar
  13. 13.
    Hess, F.: Computing Riemann–Roch spaces in algebraic function fields and related topics. J. Symb. Comput. 33(4), 425–445 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Hardouin, C., Singer, M.F.: Differential Galois theory of linear difference equations. Math. Ann. 342(2), 333–377 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Hardouin, C., Sauloy, J., Singer, M.F.: Galois Theories of Linear Difference Equations: An Introduction, Mathematical Surveys and Monographs, vol. 211. American Mathematical Society, Providence, RI, 2016, Papers from the courses held at the CIMPA Research School in Santa Marta, July 23–August 1 (2012)Google Scholar
  16. 16.
    Jordan, C.: Calculus of finite differences, 3rd edn. Introduction by Harry C. Carver. Chelsea Publishing Co., New York (1965)Google Scholar
  17. 17.
    Kurkova, I., Raschel, K.: On the functions counting walks with small steps in the quarter plane. Publ. Math. Inst. Hautes Études Sci. 116, 69–114 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Kauers, M., Yatchak, R.: Walks in the quarter plane with multiple steps. In: Proceedings of FPSAC 2015, Discrete Mathematics and Theoretical Computer Science Proceedings, Association of Discrete Mathematics and Theoretical Computer Science, Nancy, pp. 25–36 (2015)Google Scholar
  19. 19.
    Masser, D.W.: Linear Relations on Algebraic Groups, New Advances in Transcendence Theory (Durham, 1986), pp. 248–262. Cambridge University Press, Cambridge (1988)CrossRefGoogle Scholar
  20. 20.
    Melczer, S., Mishna, M.: Singularity analysis via the iterated kernel method. Comb. Probab. Comput. 23(5), 861–888 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Mishna, M., Rechnitzer, A.: Two non-holonomic lattice walks in the quarter plane. Theor. Comput. Sci. 410(38–40), 3616–3630 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Raschel, K.: Counting walks in a quadrant: a unified approach via boundary value problems. J. Eur. Math. Soc. 14(3), 749–777 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Silverman, J.H.: The Arithmetic of Elliptic Curves, 2nd edn. Springer, New York (2009). (English) Google Scholar
  24. 24.
    van der Put, M., Singer, M.F.: Galois theory of difference equations. In: Lecture Notes in Mathematics, vol. 1666. Springer, Berlin (1997)Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Thomas Dreyfus
    • 1
  • Charlotte Hardouin
    • 2
  • Julien Roques
    • 3
    Email author
  • Michael F. Singer
    • 4
  1. 1.Institut de Recherche Mathématique Avancée, U.M.R. 7501Université de Strasbourg et C.N.R.S.StrasbourgFrance
  2. 2.Institut de Mathématiques de ToulouseUniversité Paul SabatierToulouseFrance
  3. 3.Institut Fourier, CNRS UMR 5582Université Grenoble AlpesSt Martin d’HèresFrance
  4. 4.Department of MathematicsNorth Carolina State UniversityRaleighUSA

Personalised recommendations