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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Exponentiation is hard to avoid


Author: Chris Miller
Journal: Proc. Amer. Math. Soc. 122 (1994), 257-259
MSC: Primary 03C65; Secondary 03C40, 03C50, 26A12
MathSciNet review: 1195484
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Abstract: Let $ \mathcal{R}$ be an O-minimal expansion of the field of real numbers. If $ \mathcal{R}$ is not polynomially bounded, then the exponential function is definable (without parameters) in $ \mathcal{R}$. If $ \mathcal{R}$ is polynomially bounded, then for every definable function $ f:\mathbb{R} \to \mathbb{R}$, f not ultimately identically 0, there exist c, $ r \in \mathbb{R},c \ne 0$, such that $ x \mapsto {x^r}:(0, + \infty ) \to \mathbb{R}$ is definable in $ \mathcal{R}$ and $ {\lim _{x \to + \infty }}f(x)/{x^r} = c$.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1994-1195484-5
Article copyright: © Copyright 1994 American Mathematical Society