Powers are easy to avoid

Abstract

Suppose that $\tilde{\mathbb{R}}$ is an o-minimal expansion of the real field in which restricted power functions are definable. We show that if $\hat{\mathbb{R}}$ is both a reduct (in the sense of definability) of the expansion ${\tilde{\mathbb{R}}}^{\mathbb{R}}$ of $\tilde{\mathbb{R}}$ by all real power functions and an expansion (again in the sense of definability) of $\tilde{\mathbb{R}}$, then, provided that $\tilde{\mathbb{R}}$ and $\hat{\mathbb{R}}$ 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.

References in corpus (1)

• Expansions of real closed fields which introduce no new smooth functions