AN ULTRAMETRIC VERSION OF THE MAILLET-MALGRANGE THEOREM FOR NONLINEAR q-DIFFERENCE EQUATIONS

We prove an ultrametric q-difference version of the MailletMalgrange theorem on the Gevrey nature of formal solutions of nonlinear analytic q-difference equations. Since degq and ordq define two valuations on 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 |q| = 1 and a classical diophantine condition. Introduction In 1903, E. Maillet [Mai03] proved that a formal power series solution of an algebraic differential equation is Gevrey. B. Malgrange [Mal89] generalized and made more precise Maillet’s statement in the case of an analytic nonlinear differential equation. Finally C. Zhang [Zha98] proved a q-difference-differential version of the Maillet-Malgrange theorem. In the meantime a Gevrey theory for linear q-difference-differential equations has been largely developed; cf. for instance [Ram78], [Béz92b], [NM93], [FJ95]. In this paper we prove an analogue of the Maillet-Malgrange theorem for ultrametric nonlinear analytic q-difference equations, under the assumption |q| = 1. It generalizes to nonlinear q-difference equations a theorem of Bézivin and Boutabaa; cf. [BB92]. The proof follows [Mal89]. The same technique allows us to prove a Maillet-Malgrange theorem for qdifference equations when |q| = 1, both in the complex and in the ultrametric setting, under a classical diophantine hypothesis: this result generalizes the main result of [Béz92a] and answers a question asked therein. Notice that the problem of nonlinear differential equation in the ultrametric setting is treated in [SSa], [SS81], [SSb], where a p-adic avatar of diophantine conditions on the exponents is also assumed. Received by the editors November 13, 2006. 2000 Mathematics Subject Classification. Primary 33E99, 39A13. c ©2008 American Mathematical Society Reverts to public domain 28 years from publication