On the Factorization of lacunary polynomials

2022/07/24 by Filaseta, Michael
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2207.11648

Abstract

This paper addresses the factorization of polynomials of the form F(x) = f₀(x) + f₁(x) xⁿ + ⋯ + 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 ≤ j ≤ 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.

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