Powers are easy to avoid

TL;DR

This paper proves that in certain o-minimal structures, expansions by power functions do not add new definable sets if they share the same field of exponents, confirming a polynomially bounded version of a conjecture by van den Dries and Miller.

Contribution

It establishes that under specific conditions, expansions by power functions do not increase definable sets, advancing understanding of o-minimal structures and their definability properties.

Findings

Same definable sets when sharing the same field of exponents

Extension by power functions does not enlarge definable sets under conditions

Supports a polynomially bounded version of van den Dries and Miller's conjecture

Abstract

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.