Valuation theory of exponential Hardy fields I

Abstract.

We describe the valuation theoretic properties of the Hardy fields associated to models of $T(\exp)$, where T is the theory of a polynomially bounded o-minimal expansion of the reals and $\exp$ is the real exponential function. We deduce that $T(\exp)$ has levels with parameters and is exponentially bounded. We establish a maximality property of $H(\mathbb{R}_{\rm an, powers})$, the Hardy field of the expansion by the restricted analytic functions and power functions.