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 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 is at for all 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 is a polynomially bounded o-minimal expansion of the field of real numbers with restricted analytic functions, then all functions definable in are real analytic, provided that this is true for all definable functions of one variable.

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DOI:
http://dx.doi.org/10.1090/S0002-9939-1995-1257118-1

Article copyright:
© Copyright 1995
American Mathematical Society