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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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An ultrametric version of the Maillet-Malgrange theorem for nonlinear $q$-difference equations
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by Lucia Di Vizio PDF
Proc. Amer. Math. Soc. 136 (2008), 2803-2814 Request permission


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 $|q|=1$ and a classical diophantine condition.

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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
  • MR Author ID: 674036
  • Email:
  • Received by editor(s): November 13, 2006
  • Published electronically: March 21, 2008
  • Communicated by: Carmen C. Chicone
  • © Copyright 2008 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 136 (2008), 2803-2814
  • MSC (2000): Primary 33E99, 39A13
  • DOI:
  • MathSciNet review: 2399044