Exponentiation is hard to avoid

Abstract:

Let $\mathcal {R}$ be an O-minimal expansion of the field of real numbers. If $\mathcal {R}$ is not polynomially bounded, then the exponential function is definable (without parameters) in $\mathcal {R}$. If $\mathcal {R}$ is polynomially bounded, then for every definable function $f:\mathbb {R} \to \mathbb {R}$, f not ultimately identically 0, there exist c, $r \in \mathbb {R},c \ne 0$, such that $x \mapsto {x^r}:(0, + \infty ) \to \mathbb {R}$ is definable in $\mathcal {R}$ and ${\lim _{x \to + \infty }}f(x)/{x^r} = c$.