Infinite differentiability in polynomially bounded o-minimal structures

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.