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Rational normal form is a powerful tool to deal with Hamiltonian partialdifferential equations without external parameters. In this paper, we buildrational normal form with exact global control of small divisors. As anapplication to nonlinear Schr\"{o}dinger equations in Gevrey spaces, we provesub-exponentially long time stability results for generic small initial data.

Exact Global Control of Small Divisors in Rational Normal Form