Irreducibility of polynomials with a large gap

Abstract

We generalize an approach from a 1960 paper by Ljunggren, leading to a practical algorithm that determines the set of $N>\mathrm{deg}(c)+\mathrm{deg}(d)$ such that the polynomial $$f_N(x)=x^{N}c(x^{-1})+d(x)$$ is irreducible over $\mathbb{Q}$, where $c,d\in \mathbb{Z}[x]$ are polynomials with nonzero constant terms and satisfying suitable conditions. As an application, we show that $x^N-k{x}^2+1$ is irreducible for all $N\ge 5$ and $k\in \{3,4,\dots ,24\}\setminus \{9,16\}$. We also give a complete description of the factorization of polynomials of the form $x^N+k{x}^{N-1}\pm (lx+1)$ with $k,l\in \mathbb{Z}$, $k\ne l$.

25 pages, 2 figures