Parametrization of integral values of polynomials. (English) Zbl 1478.13033

Let \(k\geq 1\) and \(S\subset\mathbb{Z}^k\). We say that \(S\) is \(\mathbb{Z}\)-parametrizable (resp. Int\((\mathbb{Z})\)-parametrizable) if there exist \(p=(p_1,\ldots,p_k)\in(\mathbb{Z}[T_1,\ldots,T_m])^k\)(resp. \(f=(f_1,\ldots,f_k)\in(\text{Int}(\mathbb{Z})[T_1,\ldots,T_m])^k\) for some \(m\in\mathbb{N}\) such that \(S=p(\mathbb{Z}^m)\) (resp. \(S=f(\mathbb{Z}^m)).\) In this paper, the author study the integer-valued polynomials in one variable whose image is \(\mathbb{Z}-\)parametrizable.He is recalling a result about the classification of the polynomials in one variable with coefficients in \(\mathbb{Q}\) whose image over \(\mathbb{Z}\) is equal to the image of a polynomialwith integer coefficients. These set is generated over \(\mathbb{Z}\) by a family of polynomials \(B_\beta(X):=\frac{sX(sX-r)}{2},\) where \(\beta=\frac{r}{s}\in\mathbb{Q},\ r,s\) odd coprime integers, \(s>0.\)


13F20 Polynomial rings and ideals; rings of integer-valued polynomials
11C08 Polynomials in number theory
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