On the polynomial equivalence of subsets E and f(E) of $\Bbb Z$

1420-8938

Springer Online Journal Archives 1860-2000

Mathematics

Abstract. 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 polynomial 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 polynomial equivalence of E and f(E) implies that f is linear, but whether this is true in general for, say, $D = \Bbb Z $ is an open question.

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