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.