Defining $\mathbb{Z}$ in $\mathbb{Q}$

TL;DR

This paper demonstrates that the set of integers is definable within the rational numbers using specific first-order formulas, introduces new diophantine subsets, and discusses the limitations of existential definitions under certain conjectures.

Contribution

It provides the first universal and an $orallorall$-formula for defining ${b Z}$ in ${b Q}$, and explores diophantine properties related to quadratic extensions and non-squares.

Findings

${b Z}$ is definable in ${b Q}$ by a universal first-order formula.

An $orall ext{-}orall$-formula with one universal quantifier defines ${b Z}$ in ${b Q}$.

The set of non-squares is diophantine in ${b Q}$.

Abstract

We show that ${\mathbb Z}$ is definable in ${\mathbb Q}$ by a universal first-order formula in the language of rings. We also present an $\forall\exists$-formula for ${\mathbb Z}$ in ${\mathbb Q}$ with just one universal quantifier. We exhibit new diophantine subsets of ${\mathbb Q}$ like the complement of the image of the norm map under a quadratic extension, and we give an elementary proof of the fact that the set of non-squares is diophantine. Finally, we show that there is no existential formula for ${\mathbb Z}$ in ${\mathbb Q}$, provided one assumes a strong variant of the Bombieri-Lang Conjecture for varieties over ${\mathbb Q}$ with many ${\mathbb Q}$-rational points.