Six Lectures on Transseries, Analysable Functions and the Constructive Proof of Dulac’s Conjecture

  • Jean Écalle
Part of the NATO ASI Series book series (ASIC, volume 408)


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 ℂ v . (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”.


Critical Time Local Object Cohesive Function Small Denominator Polynomial Vector Field 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer Science+Business Media Dordrecht 1993

Authors and Affiliations

  • Jean Écalle
    • 1
  1. 1.Mathématiques, Bâtiment 425Université de Paris-SudOrsay CédexFrance

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