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 $q$ 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.