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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
Keywords:
integral-valued polynomials
Full Text:
References:
 [1] Cahen, P.-J.; Chabert, J.-L., Integer valued-polynomials, Amer. math. soc. surveys monogr., vol. 58, (1997), Amer. Math. Soc. Providence [2] Cahen, P.-J.; Chabert, J.-L., Whatʼs new about integer-valued polynomials on a subset?, (), 75-96 · Zbl 0984.13012 [3] Cahen, P.-J.; Chabert, J.-L.; Frisch, S., Interpolation domains, J. algebra, 225, 794-803, (2000) · Zbl 0990.13014 [4] Chabert, J.-L., Une caractérisation des polynômes prenant des valeurs entieres sur tous LES nombres premiers, Canad. math. bull., 99, 273-282, (1996) [5] Chabert, J.-L.; Chapman, S.T.; Smith, W.W., A basis for the ring of polynomials integer-valued on prime numbers, Lect. notes pure appl. math., 189, 271-284, (1997) · Zbl 0967.13015 [6] Frisch, S., Interpolation by integer-valued polynomials, J. algebra, 211, 562-577, (1999) · Zbl 0927.13023 [7] Gilmer, R., Sets that determine integer-valued polynomials, J. number theory, 33, 95-100, (1989) · Zbl 0695.13015 [8] McQuillan, D.L., Rings of integer-valued polynomials determined by finite sets, Math. proc. R. ir. acad., 85, 177-184, (1985) · Zbl 0596.13017 [9] McQuillan, D.L., On a theorem of R. gilmer, J. number theory, 39, 245-250, (1991) · Zbl 0739.13009 [10] Peruginelli, G.; Zannier, U., Parameterizing over $$\mathbb{Z}$$ integral values of polynomials over $$\mathbb{Q}$$, Comm. algebra, 38, 119-130, (2010) · Zbl 1219.11048
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