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On image sets of integer-valued polynomials. (English) Zbl 1239.11029
Let \(\mathrm{Int}(Z)\) be the set of all polynomials \(f\in Q[X]\) with \(f(Z)\subset Z\). Two polynomials \(f,g\in \mathrm{Int}(Z)\) are called equivalent if for some \(n\in Z\) one has either \(f(X)=g(X-n)\) or \(f(X)=g(-X-n)\). The authors show that \(f,g\in \mathrm{Int}(Z)\) satisfy \(f(Z)=g(Z)\) if and only if they are either equivalent, or for some odd \(k\) one has \(f(-X)=f(X-k)\) and \(g\) is equivalent to \(f(2X)\).

MSC:
11C08 Polynomials in number theory
13F05 Dedekind, Prüfer, Krull and Mori rings and their generalizations
13F20 Polynomial rings and ideals; rings of integer-valued polynomials
13G05 Integral domains
13B25 Polynomials over commutative rings
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