The Lagrange inversion formula on non--Archimedean fields. Non--Analytical Form of Differential and Finite Difference Equations

The classical Lagrange inversion formula is extended to analytic and non–analytic inversion problems on non–Archimedean fields. We give some applications to the field of formal Laurent series in n variables, where the non–analytic inversion formula gives explicit formal solutions of general semilinear differential and q–difference equations. We will be interested in linearization problems for germs of diffeomorphisms (Siegel center problem) and vector fields. In addition to analytic results, we give sufficient condition for the linearization to belong to some Classes of ultradifferentiable germs, closed under composition and derivation, including Gevrey Classes. We prove that Bruno’s condition is sufficient for the linearization to belong to the same Class of the germ, whereas new conditions weaker than Bruno’s one are introduced if one allows the linearization to be less regular than the germ. This generalizes to dimension n > 1 some results of [6]. Our formulation of the Lagrange inversion formula by mean of trees, allows us to point out the strong similarities existing between the two linearization problems, formulated (essentially) with the same functional equation. For analytic vector fields of C^2 we prove a quantitative estimate of a previous qualitative result of [25] and we compare it with a result of [26].