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Characterizing integers among rational numbers with a universal-existential formula

Bjorn Poonen
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Abstract

We prove that Z in definable in Q by a formula with 2 universal quantifiers followed by 7 existential quantifiers. It follows that there is no algorithm for deciding, given an algebraic family of Q-morphisms, whether there exists one that is surjective on rational points. We also give a formula, again with universal quantifiers followed by existential quantifiers, that in any number field defines the ring of integers.

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Bjorn Poonen. 2007-03-30. Characterizing integers among rational numbers with a universal-existential formula. https://doi.org/10.1353/ajm.0.0057

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