Irreducibility criteria of Schur-type and Pólya-type
type: Item
instance of: scholarly article
title: Irreducibility criteria of Schur-type and Pólya-type
zbMATH Open document ID: 1232.11112
author: Lajos Hajdu
author: R. Tijdeman
author: Kálmán Győry
published in: Monatshefte für Mathematik
publication date: 2011-08-30T00:00:00Z
review text: 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.
reviewed by: Roberto Dvornicich
Mathematics Subject Classification ID: 11R09
Mathematics Subject Classification ID: 11C08
zbMATH DE Number: 5944343
zbMATH Keywords: irreducibility
zbMATH Keywords: factors
zbMATH Keywords: polynomials
zbMATH Keywords: Schur-type
zbMATH Keywords: Pólya type
MaRDI profile type: Publication
full work available at URL: s00605-010-0241-9
OpenAlex ID: W2149407604
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DOI: 10.1007/S00605-010-0241-9
description: scientific article; zbMATH DE number 5944343