Abstract
Let \({f(x)=(x-a_1)\cdots (x-a_m)}\), where a 1, . . . , a m are distinct rational integers. In 1908 Schur raised the question whether f(x) ± 1 is irreducible over the rationals. One year later he asked whether \({(f(x))^{2^k}+1}\) is irreducible for every k ≥ 1. In 1919 Pólya proved that if \({P(x)\in\mathbb{Z}[x]}\) is of degree m and there are m rational integer values a for which 0 < |P(a)| < 2−N N! where \({N=\lceil m/2\rceil}\), then P(x) is irreducible. A great number of authors have published results of Schur-type or Pólya-type afterwards. Our paper contains various extensions, generalizations and improvements of results from the literature. To indicate some of them, in Theorem 3.1 a Pólya-type result is established when the ground ring is the ring of integers of an arbitrary imaginary quadratic number field. In Theorem 4.1 we describe the form of the factors of polynomials of the shape h(x) f(x) + c, where h(x) is a polynomial and c is a constant such that |c| is small with respect to the degree of h(x) f(x). We obtain irreducibility results for polynomials of the form g(f(x)) where g(x) is a monic irreducible polynomial of degree ≤ 3 or of CM-type. Besides elementary arguments we apply methods and results from algebraic number theory, interpolation theory and diophantine approximation.
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The authors are grateful to the referee for the useful suggestions.
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Communicated by Umberto Zannier.
To the memory of Professor E. Hlawka.
K. Győry and L. Hajdu are supported in part by the Hungarian Academy of Sciences and by the OTKA grants K67580 and K75566. L. Hajdu’s work was further supported in part by the TÁMOP 4.2.1./B-09/1/KONV-2010-0007 project. The project is implemented through the New Hungary Development Plan, co-financed by the European Social Fund and the European Regional Development Fund.
Part of the research for this paper was done in Bonn by K. Győry and R. Tijdeman as visitors of the Hausdorff Research Institute for Mathematics and the Max-Planck-Institut für Mathematik, respectively.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Győry, K., Hajdu, L. & Tijdeman, R. Irreducibility criteria of Schur-type and Pólya-type. Monatsh Math 163, 415–443 (2011). https://doi.org/10.1007/s00605-010-0241-9
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DOI: https://doi.org/10.1007/s00605-010-0241-9