Hyperserial fields


Transseries provide a universal framework for the formal asymptotics of regular solutions to ordinary differential equations at infinity. More general functional equations such as may have solutions that grow faster than any iterated exponential and thereby faster than any transseries.

In order to develop a truly universal framework for the asymptotics of regular univariate functions at infinity, we therefore need a generalization of transseries: hyperseries. Hyperexponentials and hyperlogarithms play a central role in such a program. The first non-trivial hyperexponential and hyperlogarithm are and its functional inverse , where satisfies the above equation. Formally, such functions and can be introduced for any ordinal . For instance, , , , and satisfies .

In the present work, we construct a field of hyperseries that is closed under and for all ordinals . This generalizes previous work by Schmeling in the case when , as well as the previous construction of the field of logarithmic hyperseries by van den Dries, van der Hoeven, and Kaplan.

Authors: Vincent Bagayoko, Joris van der Hoeven, Elliot Kaplan

Keywords: hyperseries, transseries, asymptotic analysis, growth scale, surreal number

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