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 ∈ Z, such that all the irreducible factors of f (a, x) are of degree ≥ 3.In this paper we perform a more systematic approach to the problem and ask about reducibility of f (a, x) with a ∈ 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 ≤ 6 and divisible by a quadratic polynomial. We also give further examples of reducible f (a, x), a ∈ Q, such that all irreducible factors are of degree ≥ 3.

On the reducibility of some special quadrinomials

“…By Eq. (2.1), we have −ap 3 + p 4 + 2apq − 3p 2 q + q 2 + a = 0, −ap 2 q + p 3 q + aq 2 − 2pq 2 + 1 = 0.…”

“…We can refer to [2,3,5,6,7,8,9,10,11,12]. A polynomial f (x) with rational coefficients is primitive reducible if it is reducible but f (x 1/l ) is not reducible for any integer l ≥ 2.…”

“…He also obtained some examples of reducible quadrinomial x n + x m + x k + a with a ∈ Z and a < −5, such that all the irreducible factors of f (a, x) are of degree ≥ 3. In 2015, A. Bremner and M. Ulas [3] studied the reducible of quadrinomials f (a, x) with 4 ≤ n ≤ 6 in a more systematic way and gave further examples of reducible f (a, x), a ∈ Q, such that all the irreducible factors with degree ≥ 3. Now we investigate the divisibility of the following quadrinomial…”