On the Factorization of lacunary polynomials

Michael Filaseta

paper

On the Factorization of lacunary polynomials

arXiv:2207.11648

Abstract

This paper addresses the factorization of polynomials of the form $F (x) = f_{0} (x) + f_{1} (x) x^{n} + \dots + f_{r - 1} (x) x^{(r - 1) n} + f_{r} (x) x^{r n}$ where $r$ is a fixed positive integer and the $f_{j} (x)$ are fixed polynomials in $Z [x]$ for $0 \leq j \leq r$ . 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 including giving two examples that arise from trace fields of hyperbolic $3$ -manifolds.