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We make a remark about an estimate of the rest for the non-resonant action-angle normal forms and exhibit a simple example suggesting the optimality of this estimate when there are no small divisors. Given a polynomial perturbation of degree P and an integer k, calling γ the size of the small denominators up to order k, we prove that the kth order remainder is bounded by (2/ε0)k+1 with ε0=const...
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