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Parametrizing over \(\mathbb Z\) integral values of polynomials over \(\mathbb Q\). (English) Zbl 1219.11048
Let \(f(X)\in {\mathbb Q}[X]\setminus {\mathbb Z}[X]\), and suppose that there exists \(g\in{\mathbb Z}[Y_1,\ldots,Y_m]\) such that \(f({\mathbb Z})=g({\mathbb Z}^m)\). The authors prove that there exist odd coprime integers \(r,s\), with \(s\) a prime power of \(1\), and a polynomial \(F\in {\mathbb Z}[X]\) such that \(f(X)=F\left(\frac{sX(r-sX)}{2}\right) \). In particular: \(2^{\frac{\deg f}{2}} f(\frac{X}{s})\in {\mathbb Z}[X]\). If \(m=1\), then \(s=1\). Such \(r,s,F\) are uniquely determined by \(f\). As a converse: let \(f(X)=F\left(\frac{sX(r-sX)}{2}\right)\) for a polynomial \(F\in {\mathbb Z}[X]\) and coprime odd integers \(r,s\), with \(s\) a prime power or \(1\); then there exists a polynomial \(g\in{\mathbb Z}[Y_1,Y_2]\) such that \(f({\mathbb Z})=g({\mathbb Z}^2)\); if either \(f\in {\mathbb Z}[X]\) or \(s=1\), there exists \(g\in {\mathbb Z}[Y]\) with \(f({\mathbb Z})=g({\mathbb Z})\). The polynomial \(f(X)\in {\mathbb Q}[X]\) is of the form \(F\left(\frac{sX(r-sX)}{2}\right)\) for an \(F\in {\mathbb Z}[X]\) and coprime odd integers \(r,s\) if and only if \(f(\frac rs - X)=f(X) \) and \(f(\frac{2X}{s})\in {\mathbb Z}[X]\).

MSC:
11C08 Polynomials in number theory
13F20 Polynomial rings and ideals; rings of integer-valued polynomials
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References:
[1] Cahen P.-J., Integer-Valued Polynomials 48 (1997)
[2] DOI: 10.1080/00927870701776938 · Zbl 1209.11038
[3] DOI: 10.1016/j.jpaa.2007.05.019 · Zbl 1215.11025
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