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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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