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

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 a...

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Bibliographic Details
Published in:arXiv.org 2008-03
Main Author: Lucia Di Vizio
Format: Article
Language:English
Subjects:
Deformation
Difference equations
Knots
Mathematical analysis
Nonlinear analysis
Nonlinear equations
Polynomials
Theorems
Online Access:Get full text
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Summary: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 ``Painleve' 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 consider also 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.
ISSN:2331-8422
DOI:10.48550/arxiv.0709.2464