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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

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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

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.
References
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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