Defining ℤ in ℚ - Koenigsmann, J (Princeton University, Department of Mathematics, 2016)
We show that Z is definable in Q by a universal first-order formula in the language of rings. We also present an ∀∃-formula for Z in Q with just one universal quantifier. We exhibit new diophantine subsets of Q like the complement of the image of the norm map under a quadratic extension, and we give an elementary proof for the fact that the set of non-squares is diophantine. Finally, we show that there is no existential formula for Z in Q, provided one assumes a strong variant of the Bombieri-Lang Conjecture for varieties over Q with many Q-rational points.