Infinite differentiability in polynomially bounded o-minimal structures

Abstract

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