Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

   
 
 

 

On the algebraic relations between Mahler functions


Author: Julien Roques
Journal: Trans. Amer. Math. Soc. 370 (2018), 321-355
MSC (2010): Primary 39A06, 12H10
DOI: https://doi.org/10.1090/tran/6945
Published electronically: July 13, 2017
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In the last years, a number of authors have studied the algebraic relations between the generating series of automatic sequences. It turns out that these series are solutions of Mahler type equations. This paper is mainly concerned with the difference Galois groups of Mahler type equations (these groups reflect the algebraic relations between the solutions of the equations). In particular, we study in detail the equations of order $ 2$ and compute the difference Galois groups of classical equations related to the Baum-Sweet and to the Rudin-Shapiro automatic sequences.


References [Enhancements On Off] (What's this?)

  • [Bec94] Paul-Georg Becker, $ k$-regular power series and Mahler-type functional equations, J. Number Theory 49 (1994), no. 3, 269-286. MR 1307967, https://doi.org/10.1006/jnth.1994.1093
  • [BM06] Mireille Bousquet-Mélou, Rational and algebraic series in combinatorial enumeration, International Congress of Mathematicians. Vol. III, Eur. Math. Soc., Zürich, 2006, pp. 789-826. MR 2275707
  • [DM82] Pierre Deligne and James S. Milne, Tannakian categories, in Hodge Cycles, Motives, and Shimura Varieties, Lecture Notes in Mathematics, vol. 900, Springer-Verlag, Berlin-New York, 1982. MR 654325
  • [Har77] Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157
  • [Hen97] Peter A. Hendriks, An algorithm for computing a standard form for second-order linear $ q$-difference equations, J. Pure Appl. Algebra 117/118 (1997), 331-352. MR 1457845, https://doi.org/10.1016/S0022-4049(97)00017-0
  • [Hen98] Peter A. Hendriks, An algorithm determining the difference Galois group of second order linear difference equations, J. Symbolic Comput. 26 (1998), no. 4, 445-461. MR 1646675, https://doi.org/10.1006/jsco.1998.0223
  • [Hur92] A. Hurwitz, Ueber algebraische Gebilde mit eindeutigen Transformationen in sich, Math. Ann. 41 (1892), no. 3, 403-442 (German). MR 1510753, https://doi.org/10.1007/BF01443420
  • [Kat87] Nicholas M. Katz, On the calculation of some differential Galois groups, Invent. Math. 87 (1987), no. 1, 13-61. MR 862711, https://doi.org/10.1007/BF01389152
  • [Kat90] Nicholas M. Katz, Exponential sums and differential equations, Annals of Mathematics Studies, vol. 124, Princeton University Press, Princeton, NJ, 1990. MR 1081536
  • [Lan52] Serge Lang, On quasi algebraic closure, Ann. of Math. (2) 55 (1952), 373-390. MR 0046388
  • [Mah30a] Kurt Mahler, Arithmetische Eigenschaften der Lösungen einer Klasse von Funktionalgleichungen, Math. Ann. 103 (1930), no. 1, 532 (German). MR 1512635, https://doi.org/10.1007/BF01455708
  • [Mah30b] Kurt Mahler, Arithmetische Eigenschaften einer Klasse transzendental-transzendenter Funktionen, Math. Z. 32 (1930), no. 1, 545-585 (German). MR 1545184, https://doi.org/10.1007/BF01194652
  • [Mah30c] Kurt Mahler, Uber das Verschwinden von Potenzreihen mehrerer Veränderlichen in speziellen Punktfolgen, Math. Ann. 103 (1930), no. 1, 573-587 (German). MR 1512638, https://doi.org/10.1007/BF01455711
  • [Nis96] Kumiko Nishioka, Mahler functions and transcendence, Lecture Notes in Mathematics, vol. 1631, Springer-Verlag, Berlin, 1996. MR 1439966
  • [NN12] Kumiko Nishioka and Seiji Nishioka, Algebraic theory of difference equations and Mahler functions, Aequationes Math. 84 (2012), no. 3, 245-259. MR 2996417, https://doi.org/10.1007/s00010-012-0132-3
  • [NvdPT08] K. A. Nguyen, M. van der Put, and J. Top, Algebraic subgroups of $ {\rm GL}_2(\mathbb{C})$, Indag. Math. (N.S.) 19 (2008), no. 2, 287-297. MR 2489331, https://doi.org/10.1016/S0019-3577(08)80004-3
  • [Pel09] F. Pellarin, An introduction to Mahler's method for transcendence and algebraic independence, in the EMS proceedings of the conference ``Hodge structures, transcendence and other motivic aspects'', G. Boeckle, D. Goss, U. Hartl, and M. Papanikolas, eds., 2009.
  • [Ph15] Patrice Philippon, Groupes de Galois et nombres automatiques, J. Lond. Math. Soc. (2) 92 (2015), no. 3, 596-614. MR 3431652, https://doi.org/10.1112/jlms/jdv056
  • [Ser68] Jean-Pierre Serre, Corps locaux, Deuxième édition; Publications de l'Université de Nancago, No. VIII, Hermann, Paris, 1968. MR 0354618
  • [vdPS97] Marius van der Put and Michael F. Singer, Galois theory of difference equations, Lecture Notes in Mathematics, vol. 1666, Springer-Verlag, Berlin, 1997. MR 1480919

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 39A06, 12H10

Retrieve articles in all journals with MSC (2010): 39A06, 12H10


Additional Information

Julien Roques
Affiliation: Institut Fourier, Université Grenoble 1, CNRS UMR 5582, 100 rue des Maths, BP 74, 38402 St. Martin d’Hères, France
Address at time of publication: Université Grenoble Alpes, Institut Fourier, CNRS UMR 5582, CS 40700, 38058 Grenoble Cedex 09, France
Email: Julien.Roques@univ-grenoble-alpes.fr

DOI: https://doi.org/10.1090/tran/6945
Keywords: Linear difference equations, difference Galois theory
Received by editor(s): April 10, 2015
Received by editor(s) in revised form: March 21, 2016
Published electronically: July 13, 2017
Article copyright: © Copyright 2017 by the author

American Mathematical Society