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 \(f(2X)\).


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
Full Text: DOI


[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.