超能力很容易避免

摘要：

设$\widetilde{\mathbb R}$是实域的一个O-极小展开式，其中限制幂函数是可定义的.我们表明，如果$\widehat{\mathbb R}$是一个减少（在可定义性的意义上）的扩展$\widetilde{\mathbb R}^{\mathbb R}$的$\widetilde{\mathbb R}$的所有实功率函数和扩展（再次在可定义性的意义上）的$\widetilde{\mathbb R}$，然后，如果$\widetilde{\mathbb R}$和$\widehat{\mathbb R}$有相同的领域的指数，他们定义相同的集合。这可以被看作是一个多项式有界版本的一个老猜想的范登德里斯和米勒。

英文摘要：

Suppose that $\widetilde{\mathbb R}$ is an o-minimal expansion of the real field in which restricted power functions are definable. We show that if $\widehat{\mathbb R}$ is both a reduct (in the sense of definability) of the expansion $\widetilde{\mathbb R}^{\mathbb R}$ of $\widetilde{\mathbb R}$ by all real power functions and an expansion (again in the sense of definability) of $\widetilde{\mathbb R}$, then, provided that $\widetilde{\mathbb R}$ and $\widehat{\mathbb R}$ have the same field of exponents, they define the same sets. This can be viewed as a polynomially bounded version of an old conjecture of van den Dries and Miller.

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