Enrico Valdinoci
A remark on sharp estimates for high order nonresonant normal
forms in Hamiltonian perturbation theory
(24K, LaTeX 2.09)

ABSTRACT.  Using the method of majorants, we give an estimate of the rest for
the nonresonant action-angle normal forms
and exhibit a simple example suggesting the ``optimality''
of this estimate. Given an integer $k$, calling $\g$
the size of the small denominators up to order $k$,
we prove that the $k^{\mbox{th}}$ order remainder
is approximatively bounded by
$O(\e_0^{-k})$ with $\e_0=O(\g^2/k)$.
Thus, if we disregard the dependence of $\g$ upon $k$, we obtain
a rest bounded by $({\mbox{const}}\;k)^k$.
These estimates are conjectured to be optimal: to support this idea we
present a simplified model problem with no small denominators,
formally related to the above calculations: this example exhibits a
factorial divergence.