Nondefinability results with entire functions of finite order in polynomially bounded o-minimal structures

Abstract

Let \({\mathcal {R}}\) be a polynomially bounded o-minimal expansion of the real field. Let f(z) be a transcendental entire function of finite order \(\rho \) and type \(\sigma \in [0,\infty ]\) . The main purpose of this paper is to show that if ( \(\rho <1\) ) or ( \(\rho =1\) and \(\sigma =0\) ), the restriction of f(z) to the real axis is not definable in \({\mathcal {R}}\) . Furthermore, we give a generalization of this result for any \(\rho \in [0,\infty )\) .