Defining $\mathbb{Z}$ in $\mathbb{Q}$ - MathOverflow most recent 30 from http://mathoverflow.net2013-05-20T07:48:07Zhttp://mathoverflow.net/feeds/question/86861http://www.creativecommons.org/licenses/by-nc/2.5/rdfhttp://mathoverflow.net/questions/86861/defining-mathbbz-in-mathbbqDefining $\mathbb{Z}$ in $\mathbb{Q}$Math-player2012-01-27T21:17:05Z2012-01-28T08:24:04Z
<p>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?</p>
<p>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: <a href="http://arxiv.org/abs/1011.3424" rel="nofollow">http://arxiv.org/abs/1011.3424</a> (Koenigsmann's
paper), and <a href="http://www-math.mit.edu/~poonen/papers/ae.pdf" rel="nofollow">http://www-math.mit.edu/~poonen/papers/ae.pdf</a> (Poonen's paper).</p>
<p>Thank you</p>