2010/11/15 by Koenigsmann, Jochen
#11R35 #11U09 #FOS: Mathematics #Logic (math.LO) #Number Theory (math.NT) #Primary: 11U05 #Secondary: 11R52
paper · doi:10.48550/arxiv.1011.3424
We show that \mathbb Z is definable in \mathbb Q by a universal first-order formula in the language of rings. We also present an ∀∃-formula for \mathbb Z in \mathbb Q with just one universal quantifier. We exhibit new diophantine subsets of \mathbb Q like the complement of the image of the norm map under a quadratic extension, and we give an elementary proof of the fact that the set of non-squares is diophantine. Finally, we show that there is no existential formula for \mathbb Z in \mathbb Q, provided one assumes a strong variant of the Bombieri-Lang Conjecture for varieties over \mathbb Q with many \mathbb Q-rational points.