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Proceedings of the American Mathematical Society

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Infinite differentiability in polynomially bounded o-minimal structures

Author: Chris Miller
Journal: Proc. Amer. Math. Soc. 123 (1995), 2551-2555
MSC: Primary 03C65; Secondary 03C50, 26E10
MathSciNet review: 1257118
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Abstract: Infinitely differentiable functions definable in a polynomially bounded o-minimal expansion $ \Re $ of the ordered field of real numbers are shown to have some of the nice properties of real analytic functions. In particular, if a definable function $ f:{\mathbb{R}^n} \to \mathbb{R}$ is $ {C^N}$ at $ a \in {\mathbb{R}^n}$ for all $ N \in \mathbb{N}$ and all partial derivatives of f vanish at a, then f vanishes identically on some open neighborhood of a. Combining this with the Abhyankar-Moh theorem on convergence of power series, it is shown that if $ \Re $ is a polynomially bounded o-minimal expansion of the field of real numbers with restricted analytic functions, then all $ {C^\infty }$ functions definable in $ \Re $ are real analytic, provided that this is true for all definable functions of one variable.

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Article copyright: © Copyright 1995 American Mathematical Society

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