We show that every polynomial overring of the ring Int(Z) of polynomials which are integer-valued over Z may be considered as the ring of polynomials which are integer-valued over some subset of Ẑ, the profinite completion of Z with respect to the fundamental system of neighbourhoods of 0 consisting of all non-zero ideals of Z.

If, for a subset S of Z, we compare the conditions of being parametrizable (a) by a single k-tuple of polynomials with integer coefficients, (b) by a single k-tuple of integer-valued polynomials and (c) by finitely many k-tuples of polynomials with integer coefficients (variables ranging through the integers in each case), then a ⇒ b (obviously), b ⇒ c, and neither implication is reversible. We...