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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 (Definability and decision problems in arithmetic). The references for the latest results are: Defining $\mathbb{Z}$ in $\mathbb{Q}$ (Koenigsmann's paper, arXiv version), and Characterizing integers among rational numbers with a universal-existential formula (Poonen's paper, arXiv version).

Thank you

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). The references for the latest results are: Defining $\mathbb{Z}$ in $\mathbb{Q}$ (Koenigsmann's paper, arXiv version), and Characterizing integers among rational numbers with a universal-existential formula (Poonen's paper, arXiv version).

Thank you