Exponentiation is hard to avoid

Source abstract

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