On the Factorization of lacunary polynomials

TL;DR

This paper presents an efficient method to analyze the factorization of lacunary polynomials with large exponents, showing that under certain conditions, their non-reciprocal parts are trivial or irreducible, with applications to hyperbolic 3-manifolds.

Contribution

It introduces a novel approach for factorization of lacunary polynomials with large exponents, focusing on the non-reciprocal part and its irreducibility.

Findings

Non-reciprocal part of the polynomial is either 1 or irreducible for large n.

Method applies to polynomials arising from trace fields of hyperbolic 3-manifolds.

Provides examples illustrating the approach and its applications.

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.