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Irreducibility of polynomials with a large gap

Volume 192 / 2020

Will Sawin, Mark Shusterman, Michael Stoll Acta Arithmetica 192 (2020), 111-139 MSC: 11R09, 12E05, 11C08, 13P05. DOI: 10.4064/aa180526-12-6 Published online: 8 November 2019

Abstract

We generalize an approach from a 1960 paper by Ljunggren, leading to a practical algorithm that determines the set of exponents $N \gt \deg c + \deg d$ such that the polynomial \[ f_N(x) = x^N c(x^{-1}) + d(x) \] is irreducible over $\mathbb Q $, where $c, d \in \mathbb Z [x]$ are polynomials with non-zero constant terms and satisfying suitable conditions. As an application, we show that $x^N - k x^2 + 1$ is irreducible for all $N \ge 5$, for every $k \in \{3, 4, \ldots , 24\} \setminus \{9, 16\}$. We also give a complete description of the factorization of polynomials of the form $x^N + k x^{N-1} \pm (l x + 1)$ with $k, l \in \mathbb Z $, $k \neq l$.

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