## 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, $$s>0.$$

### MSC:

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

### Keywords:

integer-valued polynomials
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### References:

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