On the IV-equivalence of subsets E and f(E) of Z
R. Gilmer, W. W. Smith
For a subset E of an integral domain D and an integer-valued polynomial f over D, we investigate conditions under which the subsets E and f(E) of D determine the same integer-valued polynomials on D (this is the definition of IV-equivalence of E and f(E)). Our primary interest in this problem lies in the case where D is the ring of rational integers. Using work of McQuillan, the case where E is finite is resolved completely in Section 3. For E infinite we show in several cases that IV-equivalence of E and f(E) implies that f is linear, but whether this is true in general for, say, D=Z is an open question.