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

TL;DR

This paper establishes an ultrametric version of the Maillet-Malgrange theorem for nonlinear q-difference equations, analyzing the Gevrey nature and growth of formal solutions with applications to q-deformations and knot invariants.

Contribution

It introduces an ultrametric framework for the Maillet-Malgrange theorem, extending it to nonlinear q-difference equations and exploring growth properties of solutions.

Findings

Proves an ultrametric Gevrey theorem for nonlinear q-difference equations.

Analyzes growth of degree and order of solutions in q.

Provides examples including q-Painleve II and colored Jones polynomials.

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.