Some observations concerning reducibility of quadrinomials

Abstract

In a recent paper \cite{Jan}, 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 $\ge 3$. In this paper we perform a more systematic approach to the problem and ask about reducibility of $f(a,x)$ with $a\in\Q$. 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 $\le 6$ and divisible by a quadratic polynomial. We also give further examples of reducible $f(a,x)$, $a\in\Q$, such that all irreducible factors are of degree $\ge 3$.

23 pages, to appear in Pariodica Math. Hungarica