It was proved by Poonen that $\mathbb{Z}$ is definable in $\mathbb{Q}$ 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. The references for the latest results are: http://arxiv.org/abs/1011.3424 (Koenigsmann's paper), and http://www-math.mit.edu/~poonen/papers/ae.pdf (Poonen's paper).

Thank you