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

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 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.
Document type :
Journal articles
Domain :

https://hal.archives-ouvertes.fr/hal-00350715
Contributor : Lucia Di Vizio <>
Submitted on : Wednesday, January 7, 2009 - 3:29:19 PM
Last modification on : Friday, April 10, 2020 - 5:19:25 PM

### Identifiers

• HAL Id : hal-00350715, version 1
• ARXIV : 0709.2464

### Citation

Lucia Di Vizio. An ultrametric version of the Maillet-Malgrange theorem for nonlinear q-difference equations. Proceedings of the American Mathematical Society, American Mathematical Society, 2008, 136 (1), pp.2803-2814. ⟨hal-00350715⟩

Record views