Characterizing integers among rational numbers with a universal-existential formula

We prove that ${\Bbb Z}$ in definable in ${\Bbb Q}$ by a formula with two universal quantifiers followed by seven existential quantifiers. It follows that there is no algorithm for deciding, given an algebraic family of ${\Bbb Q}$-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.