Abstract
The present paper gives a rapid, self-contained introduction to some new resummotion methods, which are noticeable for their high content in structure and revolve logically around the notions of resurgence, compensation and acceleration. Then it presents three applications of decreasing generality: (A) The study of analytic singularities and local objects, mainly singular analytic vector fields and local diffeomorphisms of ℂv. (B) The construction of the fields of transseries and analysable germs, the latter being essentially the broadest extension of the ring of real-analytic germs whose elements tolerate all common operations, including integration, and yet retain the property of being wholly formalizable, i.e. reducible to a properly structured set of real coefficients. (C) The proof of the non-accumulation of limit-cycles for real-analytic, first-order differential equations.
Each of the topics selected for inclusion in this survey is closely related to the rest, with one red thread running through everything, namely the Analytic Principle, which “posits” that local entities arising naturally out of a local analytic situation can be entirely “formalized”.
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Écalle, J. (1993). Six Lectures on Transseries, Analysable Functions and the Constructive Proof of Dulac’s Conjecture. In: Schlomiuk, D. (eds) Bifurcations and Periodic Orbits of Vector Fields. NATO ASI Series, vol 408. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-8238-4_3
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DOI: https://doi.org/10.1007/978-94-015-8238-4_3
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