  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.
