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An ultrametric version of the Maillet-Malgrange theorem for nonlinear $ q$-difference equations


Author: Lucia Di Vizio
Journal: Proc. Amer. Math. Soc. 136 (2008), 2803-2814
MSC (2000): Primary 33E99, 39A13
DOI: https://doi.org/10.1090/S0002-9939-08-09352-0
Published electronically: March 21, 2008
MathSciNet review: 2399044
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Abstract: We prove an ultrametric $ q$-difference version of the Maillet- Malgrange theorem on the Gevrey nature of formal solutions of nonlinear analytic $ q$-difference equations. Since $ \deg_q$ and $ {ord}_q$ define two valuations on $ \mathbb{C}(q)$, we obtain, in particular, a result on the growth of the degree in $ q$ and the order at $ q$ of formal solutions of nonlinear $ q$-difference equations, when $ q$ is a parameter. We illustrate the main theorem by considering two examples: a $ q$-deformation of ``Painlevé II'', for the nonlinear situation, and a $ q$-difference equation satisfied by the colored Jones polynomials of the figure $ 8$ knots, in the linear case.

We also consider a $ q$-analog of the Maillet-Malgrange theorem, both in the complex and in the ultrametric setting, under the assumption that $ \vert q\vert=1$ and a classical diophantine condition.


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

  • [A'C69] Norbert A'Campo.
    Théorème de préparation différentiable ultra-métrique.
    In Séminaire Delange-Pisot-Poitou: 1967/68, Théorie des Nombres, Fasc. 2, Exp. 17. Secrétariat mathématique, Paris, 1969. MR 0244240 (39:5557)
  • [BB92] Jean-Paul Bézivin and Abdelbaki Boutabaa.
    Sur les équations fonctionelles $ p$-adiques aux $ q$-différences.
    Universitat de Barcelona. Collectanea Mathematica, 43(2):125-140, 1992. MR 1223416 (95j:39031)
  • [Béz92a] Jean-Paul Bézivin.
    Convergence des solutions formelles de certaines équations fonctionnelles.
    Aequationes Mathematicae, 44(1):84-99, 1992. MR 1165786 (93d:39011)
  • [Béz92b] Jean-Paul Bézivin.
    Sur les équations fonctionnelles aux $ q$-différences.
    Aequationes Mathematicae, 43(2-3):159-176, 1992. MR 1158724 (93m:39006)
  • [CCG$ ^+$94] D. Cooper, M. Culler, H. Gillet, D. D. Long, and P. B. Shalen.
    Plane curves associated to character varieties of $ 3$-manifolds.
    Inventiones Mathematicae, 118(1):47-84, 1994. MR 1288467 (95g:57029)
  • [DV02] Lucia Di Vizio.
    Arithmetic theory of $ q$-difference equations. The $ q$-analogue of Grothendieck-Katz's conjecture on $ p$-curvatures.
    Inventiones Mathematicae, 150(3):517-578, 2002. MR 1946552 (2005a:12013)
  • [DVz] Lucia Di Vizio. Local analytic classification of $ q$-difference equations with $ \vert q\vert=1$. arXiv:0802.4223
  • [DVRSZ03] L. Di Vizio, J.-P. Ramis, J. Sauloy, and C. Zhang.
    Équations aux $ q$-différences.
    Gazette des Mathématiciens, (96):20-49, 2003. MR 1988639 (2004e:39023)
  • [FJ95] Monique Fleinert-Jensen.
    Théorèmes d'indices précisés et convergences des solutions pour une équation linéaire aux $ q$-différences.
    Comptes Rendus de l'Académie des Sciences. Série I. Mathématique, 321(4):425-428, 1995. MR 1351090 (96j:39004)
  • [Gar04] Stavros Garoufalidis.
    On the characteristic and deformation varieties of a knot.
    Geom. Topol. Monogr., 7:291-309, 2004. MR 2172488 (2006j:57028)
  • [KMN$ ^+$05] K. Kajiwara, T. Masuda, M. Noumi, Y. Ohta, and Y. Yamada.
    Construction of hypergeometric solutions to the $ q$-Painlevé equations.
    International Mathematics Research Notices, (24):1441-1463, 2005. MR 2153786 (2006e:33027)
  • [Mai03] Edmond Maillet.
    Sur les séries divergentes et les équations différentielles.
    Annales Scientifiques de l'École Normale Supérieure. Troisième Série, 20, 1903.
  • [Mal89] Bernard Malgrange.
    Sur le théorème de Maillet.
    Asymptotic Analysis, 2(1):1-4, 1989. MR 991413 (90f:32005)
  • [NM93] Fabienne Naegele (Marotte).
    Théorèmes d'indices pour les équations $ q$-différences-différentielles.
    Comptes Rendus de l'Académie des Sciences. Série I. Mathématique, 317(6):579-582, 1993. MR 1240803 (94h:34021)
  • [Ram78] J.-P. Ramis.
    Dévissage Gevrey.
    In Journées Singulières de Dijon (Univ. Dijon, Dijon, 1978), volume 59 of Astérisque, pages 4, 173-204. Soc. Math. France, Paris, 1978. MR 542737 (81g:34010)
  • [RGTT01] A. Ramani, B. Grammaticos, T. Tamizhmani, and K. M. Tamizhmani.
    Special function solutions of the discrete Painlevé equations.
    Computers & Mathematics with Applications. An International Journal, 42(3-5):603-614, 2001.
    Advances in difference equations, III. MR 1838017 (2002e:33033)
  • [Ser06] Jean-Pierre Serre.
    Lie algebras and Lie groups, volume 1500 of Lecture Notes in Mathematics.
    Springer-Verlag, Berlin, 2006. MR 2179691 (2006e:17001)
  • [SSa] Yasutaka Sibuya and Steven Sperber.
    Convergence of power series solutions of $ p$-adic nonlinear differential equation.
    In Recent advances in differential equations (Trieste, 1978), pages 405-419. MR 643150 (83f:12021)
  • [SSb] Yasutaka Sibuya and Steven Sperber.
    Some new results on power-series solutions of algebraic differential equations.
    In Singular perturbations and asymptotics (Proc. Adv. Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1980), volume 45 of Publ. Math. Res. Center Univ. Wisconsin, pages 379-404. MR 606047 (82d:34072)
  • [SS81] Yasutaka Sibuya and Steven Sperber.
    Arithmetic properties of power series solutions of algebraic differential equations.
    Annals of Mathematics. Second Series, 113:111-157, 1981. MR 604044 (82j:12022)
  • [Zha98] Changgui Zhang.
    Sur un théorème du type de Maillet-Malgrange pour les équations $ q$-différences-différentielles.
    Asymptotic Analysis, 17(4):309-314, 1998. MR 1656811 (99j:35005)

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Additional Information

Lucia Di Vizio
Affiliation: Institut de Mathématiques de Jussieu, Topologie et géométrie algébriques, Case 7012, 2, place Jussieu, 75251 Paris Cedex 05, France
Email: divizio@math.jussieu.fr

DOI: https://doi.org/10.1090/S0002-9939-08-09352-0
Received by editor(s): November 13, 2006
Published electronically: March 21, 2008
Communicated by: Carmen C. Chicone
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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