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.