Let Int(\(\mathbb{Z}\))\(=\{f\in\mathbb{Q}[X] \mid f(\mathbb{Z})\subset\mathbb{Z}\}\) be the ring of integer-valued polynomials. It is well-known that this ring is not a unique factorization domain and its elasticity, defined as the supremum of the set of ratios of lengths of two decompositions of irreducible factors of nonzero nonunit elements, is infinite [P.-J. Cahen and J.-L. Chabert, J. Pure Appl. Algebra 103, No. 3, 303–311 (1995; Zbl 0843.12001)]. Moreover, Int(\(\mathbb{Z}\)) is fully elastic, which means that every rational number greater than \(1\) occurs as the elasticity of some nonzero nonunit element ([S. T. Chapman and B. A. McClain, J. Algebra 293, No. 2, 595–610 (2005; Zbl 1082.13001)]).
This paper proves again these last two results in a constructive way and goes beyond that. The main result is the following: given a finite set \(S=\{n_1,\dots,n_r\}\) of positive integers greater than \(1\), there exists a polynomial \(f\in\)Int(\(\mathbb{Z}\)) such that \(f(X)\) admits \(r\) distinct factorizations into irreducibles of length \(n_1,\dots,n_r\), respectively. The proof is constructive and allows multiplicities of lengths of factorizations to be specified. Moreover, the common denominator of the coefficients of the polynomial \(f(X)\) is a prime \(p\), that is, \(f(X)\) is of the form \(\frac{g(X)}{p}\), for some polynomial \(g(X)\) with integer coefficients.
The proof is obtained by recalling first the possible factorizations that an integer-valued polynomial of the form \(\frac{g(X)}{p}\) can have inside Int(\(\mathbb{Z}\)), which depends on the irreducible factors of \(g(X)\) in \(\mathbb{Z}[X]\). Then, the core of the above main result is Lemma 6 which shows the following: given a finite number of polynomials \(f_1(X),\dots,f_n(X)\) in \(\mathbb{Z}[X]\), there exist irreducible polynomials \(F_1(X),\dots,F_n(X)\) in \(\mathbb{Z}[X]\), pairwise non-associated, with \(\deg(f_i)=\deg(F_i)\) for all \(i=1,\dots,n\), such that the fixed divisor of the product of a subset of the \(f_i(X)\)’s is the same as the fixed divisor of the product of the \(F_i(X)\)’s.