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Hyperserial fields

Abstract : Transseries provide a universal framework for the formal asymptotics of regular solutions to ordinary differential equations at infinity. More general functional equations such as E (x + 1) = exp E (x) 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 E and its functional inverse L, where E satisfies the above equation. Formally, such functions E and L can be introduced for any ordinal.
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https://hal.archives-ouvertes.fr/hal-03196388
Contributor : Joris Van Der Hoeven Connect in order to contact the contributor
Submitted on : Monday, April 12, 2021 - 5:42:10 PM
Last modification on : Thursday, April 15, 2021 - 3:30:06 AM
Long-term archiving on: : Tuesday, July 13, 2021 - 7:10:57 PM

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  • HAL Id : hal-03196388, version 1

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Vincent Bagayoko, Joris van Der Hoeven, Elliot Kaplan. Hyperserial fields. 2021. ⟨hal-03196388⟩

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