Some observations concerning reducibility of quadrinomials

Abstract

In a recent paper [4], Jankauskas proved some interesting results concerning the reducibility of quadrinomials of the form f (4, x), where ${f (a, x) = x^n + x^m + x^k + a}$. He also obtained some examples of reducible quadrinomials f (a, x) with ${a \in \mathbb{Z}}$, such that all the irreducible factors of f (a, x) are of degree ${\geqq 3}$.

In this paper we perform a more systematic approach to the problem and ask about reducibility of f (a, x) with ${a \in \mathbb{Q}}$. In particular by computing the set of rational points on some genus two curves we characterize in several cases all quadrinomials f (a, x) with degree ${\leqq 6}$ and divisible by a quadratic polynomial. We also give further examples of reducible ${f (a, x), a \in \mathbb{Q}}$, such that all irreducible factors are of degree ${\geqq 3}$.