It was proved by Poonen that $\mathbb{Z}$ is definable in the structure $(\mathbb{Q}, +, \cdot, 0, 1)$ using $\forall \exists$ formula. Koenigsmann has shown that $\mathbb{Z}$ is in fact definable by universal formula. What is the simplest geometric interpretation of these results?

EDIT: It is important to note, as Joel says, that the first result in this direction was that of Julia Robinson in 1948 ([Definability and decision problems in arithmetic](https://doi.org/10.2307/2266510)).
The references for the latest results are: _[Defining $\mathbb{Z}$ in $\mathbb{Q}$](http://dx.doi.org/10.4007/annals.2016.183.1.2)_ (Koenigsmann's
 paper, [arXiv version](http://arxiv.org/abs/1011.3424)), and _[Characterizing integers among rational numbers with a universal-existential formula](https://doi.org/10.1353/ajm.0.0057)_ (Poonen's paper, [arXiv version](https://arxiv.org/abs/math/0703907)).

 Thank you