Peruginelli, Giulio Parametrization of integral values of polynomials. (English) Zbl 1478.13033 Actes Rencontres C.I.R.M. 2, No. 2, 41-49 (2010). 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, \(s>0.\) Reviewer: Cristodor-Paul Ionescu (Bucureşti) MSC: 13F20 Polynomial rings and ideals; rings of integer-valued polynomials 11C08 Polynomials in number theory Keywords:integer-valued polynomials × Cite Format Result Cite Review PDF Full Text: DOI References: [1] P.-J. Cahen and J.-L. Chabert, \(Integer-Valued Polynomials\), Amer. Math. Soc. Surveys and Monographs, 48, Providence, 1997. · Zbl 0884.13010 [2] S. Frisch, \(Remarks on polynomial parametrization of sets of integer points\), Comm. Algebra 36 (2008), no. 3, 1110-1114. · Zbl 1209.11038 [3] S. Frisch, L. Vaserstein, \(Parametrization of Pythagorean triples by a single triple of polynomials\), Pure Appl. Algebra 212 (2008), no. 1, 271-274. · Zbl 1215.11025 [4] G. Peruginelli, U. Zannier, \(Parametrizing over \)\Bbb Z\( integral values of polynomials over \)\Bbb Q\[, Comm. Algebra 38 (2010), no. 1, 119-130. | Copyright Cellule MathDoc 20\] · 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.