Irreducibility criteria of Schur-type and Pólya-type
The paper addresses the question of whether some polynomials with integer coefficients are irreducible. The classical reference examples, for which irreduciblity has been proved, are \(P(x)=f(x)\pm 1\), where \(f(x)=(x-a_1)\cdots(x-a_m)\) for some integers \(a_1,\dots,a_m\) (Schur) and \(P(x)\) of degree \(m\) for which there exist \(m\) integer values \(a\) with \(0<|P(a)|<2^{-N}N!\), where \(N=\lceil{\frac m2}\rceil\) (Pólya). These old results have been generalized in various ways. Here the authors call the polynomials of Schur-type if they are of the form \(P(x)=g(h(x)f(x))\) where \(f(x),g(x),h(x)\in{\mathbb Z}[x]\), \(f(x)\) has simple roots and \(g(x)\) is irreducible; they call the polynomials of Pólya-type if they take on non-zero, but small absolute values, at many integer points. The theorems of the paper are of the following style: -- all polynomials of a certain kind are irreducible; -- all polynomials of a certain kind to not have factors of degree \(k\) for some natural number \(k\); -- all polynomials of a certain kind are irreducible except those from a given explicit list. The tools for the proofs are mainly elementary, but now and then they involve some techniques coming from algebraic number theory and diophantine approximation.
scientific article; zbMATH DE number 5944343
scientific article