Transseries as germs of surreal functions

Berarducci, Alessandro; Mantova, Vincenzo

doi:10.1090/tran/7428

Transseries as germs of surreal functions
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by Alessandro Berarducci and Vincenzo Mantova PDF

Trans. Amer. Math. Soc. 371 (2019), 3549-3592 Request permission

Abstract:

We show that Écalle’s transseries and their variants (LE and EL-series) can be interpreted as functions from positive infinite surreal numbers to surreal numbers. The same holds for a much larger class of formal series, here called omega-series. Omega-series are the smallest subfield of the surreal numbers containing the reals, the ordinal omega, and closed under the exp and log functions and all possible infinite sums. They form a proper class, can be composed and differentiated, and are surreal analytic. The surreal numbers themselves can be interpreted as a large field of transseries containing the omega-series, but, unlike omega-series, they lack a composition operator compatible with the derivation introduced by the authors in an earlier paper.

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