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We will show how to construct complex fields of transseries , which contain , an infinitely large variable , and which are closed under the field operations, infinite summation, exponential and logarithm. Fields of complex transseries are valued differential fields which have various interesting closure properties: any differentially algebraic equation over admits a solution in , and is Picard-Vessiot closed. An interesting problem is to generalize Écalle's accelero-summation theory to complex transseries. We also give an application to the problem of zero-testing for certain differentially algebraic power series.

**Occasions:** Luminy, November 21

**Documents:** slideshow, TeXmacs
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