On the factorization of lacunary polynomials

Abstract

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 including giving two examples that arise from trace fields of hyperbolic $3$-manifolds.