Infinite Differentiability in Polynomially Bounded O-Minimal Structures

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 <!-- MATH $f:{\mathbb{R}^n} \to \mathbb{R}$ --> is at <!-- MATH $a \in {\mathbb{R}^n}$ --> for all <!-- MATH $N \in \mathbb{N}$ --> 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 is a polynomially bounded o-minimal expansion of the field of real numbers with restricted analytic functions, then all <!-- MATH ${C^\infty }$ --> functions definable in are real analytic, provided that this is true for all definable functions of one variable.