Some nondefinability results with entire functions in a polynomially bounded o-minimal structure

Abstract

Let $f(z)=\Sigma _{k\ge 0}a_{k}z^{k}$ be a transcendental entire function with real coefficients. The main purpose of this paper is to show that the restriction of f to $\mathbb {R}$ is not definable in the ordered field of real numbers with restricted analytic functions, $\mathbb {R}_{\text {an}}$. Furthermore, we show that there is $\theta \in \mathbb {R}$ such that the function $f(xe^{i\theta })$ on $\mathbb {R}$ is not definable in $\mathbb {R}_{\mathcal {G}}$, where $\mathbb {R}_{\mathcal {G}}$ the expansion of the real field generated by multisummable real series.