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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

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Do you have a reference for these results? (without knowing how this is proved, it will be difficult to give a geometric interpretation) –  Guillaume Brunerie Jan 27 '12 at 23:01
See this question: mathoverflow.net/questions/19840/… @Guillaume: Here is a link to Poonen's paper lifted directly from above www-math.mit.edu/~poonen/papers/ae.pdf –  Zack Wolske Jan 27 '12 at 23:22
Probably it should be mentioned in the question that the first known definition of $\mathbb{Z}$ in $\mathbb{Q}$, a very surprising at the time, was the 1948 result of Julia Robinson. Poonen's impressive result should be seen as a refinement of Robinson's theorem, lowering the complexity of the definition. –  Joel David Hamkins Jan 28 '12 at 0:42

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