Study of Diophantine phenomena and exact asymptotic analysis of singular partial differential equations

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We showed analytical methods without the Diophantine condition in treating divergence of formal transformations in the normal form theory of a vector field. Our methods, based on asymptotic analysis, extend the so-called Borel-Laplace method, and give a real meaning to a divergent series in certain subdomain. Moreover, we give a meaning, by a Borel resummation method, to divergent first integrals of Hamiltonian systems , for which analytic non-integrability and smooth integrability occur similtaneously.