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On the factorization of lacunary polynomials

Volume 210 / 2023

Michael Filaseta Acta Arithmetica 210 (2023), 23-52 MSC: Primary 11R09; Secondary 11C08, 12E05, 57M50. DOI: 10.4064/aa220723-16-5 Published online: 5 July 2023


This paper addresses the factorization of polynomials of the form $F(x) = f_{0}(x) + f_{1}(x) x^{n} + \cdots + f_{r-1}(x) x^{(r-1)n} + f_{r}(x) x^{rn}$ where $r$ is a fixed positive integer and the $f_{j}(x)$ are fixed polynomials in $\mathbb Z[x]$ for $0 \le j \le r$. We provide an efficient method for showing that for $n$ sufficiently large and reasonable conditions on the $f_{j}(x)$, the non-reciprocal part of $F(x)$ is either $1$ or irreducible. We illustrate the approach with a few examples, including two examples that arise from trace fields of hyperbolic $3$-manifolds.


  • Michael FilasetaMathematics Department
    University of South Carolina
    Columbia, SC 29208, USA

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