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 \(\rho \in [0,\infty )\) .

中文翻译：

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多项式有界 o 极小结构中有限阶整个函数的不可定义性结果

摘要

令\({\mathcal {R}}\)为实数域的多项式有界 o 最小展开式。设f ( z ) 为有限阶\(\rho \)且类型为\(\sigma \in [0,\infty ]\)的超越整函数。本文的主要目的是证明如果 ( \(\rho <1\) ) 或 ( \(\rho =1\) and \(\sigma =0\) )， f ( z )的限制为实轴在\({\mathcal {R}}\)中不可定义。此外，我们对任何\(\rho \in [0,\infty )\)给出了该结果的概括。