## Infinite differentiability in polynomially bounded o-minimal structures

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- by Chris Miller
- Proc. Amer. Math. Soc.
**123**(1995), 2551-2555 - DOI: https://doi.org/10.1090/S0002-9939-1995-1257118-1
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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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## Bibliographic Information

- © Copyright 1995 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**123**(1995), 2551-2555 - MSC: Primary 03C65; Secondary 03C50, 26E10
- DOI: https://doi.org/10.1090/S0002-9939-1995-1257118-1
- MathSciNet review: 1257118