Abstract
In this article, we give a normal form for real-analytic, Levi-nondegenerate submanifolds of
Funding statement: Research of the authors was supported by ANR grant “ANR-14-CE34-0002-01” and the FWF (Austrian Science Foundation) grant I1776 for the international cooperation project “Dynamics and CR geometry”. Research of Bernhard Lamel was also supported by the Qatar Science Foundation, NPRP 7-511-1-098.
A Computations
We recall that
To obtain
We have
The function
We have, for
Hence, the
with
Let us first compute
In the sums below, the terms appear with some positive multiplicity that we do not write since we are only interested in a lower bound of vanishing order of the terms. From these computations, we easily obtain
Let us set as notation
B Big Denominators theorem for nonlinear systems of PDEs
In this section we recall one of the main results of article [24] about local analytic solvability of some nonlinear systems of PDEs that have the “Big Denominators property”.
B.1 The problem
Let
Given
Definition 9.
A map
such that
Denote by
the local coordinates in
Definition 10.
Let q be a nonnegative integer.
Let
We shall say that it increases the order at the origin (respectively strictly) by q if for all
(respectively
Assume that
as in Definition 9. We call it regular if, for any formal map
where
(B.1)We have set
Let us consider linear maps
Let us consider a differential analytic map of order
We consider the equation
In [24], we gave a sufficient condition on the triple
B.2 Big Denominators. Main theorem
We can define the Big Denominators property of the triple
Definition 11.
The triple of maps
The linear map
The map
(B.3)where
Remark 12.
Let
where the sum is taken over all
or the modified Fisher–Belitskii norm
Remark 13.
In practice, for each i, there is a decomposition into direct sums
with
Theorem 14 ([24, Theorem 7]).
Let us consider a system of analytic nonlinear PDEs such as equation (B.2):
If the triple
Remark 15.
The precise statement of [24, Theorem 7] holds for
B.3 Application
In this subsection we shall devise the strictly increasing condition in more detail. We look for
a formal solution
Here
Indeed, we would have
We emphasize that condition (B.5) just means that W strictly increases the order by q as defined in Definition 10. Let us look closer to that condition. Let us denote
Let us Taylor expand
We recall that
In order that the first derivative part of this Taylor expansion satisfies (B.5), it is sufficient that
This is nothing but the regularity condition as defined in Definition 10. Let us consider the other terms in the Taylor expansion. We have, for instance,
If
Corollary 16.
If
Acknowledgements
We thank the referees for their very careful reading that helped us improve our text.
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