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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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
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.


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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: http://dx.doi.org/10.1090/S0002-9939-08-09352-0
PII: S 0002-9939(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.