01-370 Timoteo Carletti

The Lagrange inversion formula on non--Archimedean fields. Non--Analytical Form of Differential and Finite Difference Equations. (105K, AMS-Latex 2e) Oct 12, 01

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Abstract. The classical formula of Lagrange for the inversion of analytic functions 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. In particular we will be interested in studying the Siegel center problem (Linearization of germs of functions near a fixed point) and the problem of conjugation of a vector fields with its linear part near a critical point. In addition to the analytic Siegel Center problem, 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 the Bruno condition is sufficient for the linearization to belong to the same Class of the germ. If one allows the linearization to be less regular than the germ new conditions are introduced, weaker than the Bruno condition. This generalizes to dimension $n\geq 1$ some results of~\cite{CarlettiMarmi}. A similar statement holds in the case of the linearization of ultradifferentiable vector fields. Moreover our formulation of the non--analytic Lagrange inversion formula by mean of the tree formalism, allows us to point out the strong similarities existing between the two linearization problems, formulated (essentially) with the same functional equation. In the case of analytic vector fields of $\C^2$ we prove a quantitative estimate of a previous qualitative result of~\cite{YoccozPerezMarco}.

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