 01370 Timoteo Carletti
 The Lagrange inversion formula on nonArchimedean fields.
NonAnalytical Form of Differential and Finite Difference Equations.
(105K, AMSLatex 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 nonanalytic inversion problems
on nonArchimedean fields. We give some applications to the field
of formal Laurent series in $n$ variables, where the nonanalytic
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 nonanalytic 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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