On the reducibility of certain quadrinomials

Some observations concerning reducibility of quadrinomials

“…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.…”

“…Tying now all the results together we get the statement of the Theorem. 4. Reducible quadrinomials f n,m,k (a, x) with deg f = 6…”

“…Case (m, k) = (5,4). Solving the equation F 6,5,4 (p, q) = 0 in rational numbers p, q leads to p 2 (4p 4 − 8p 3 + 9p 2 − 8p + 4) = , so leads to p = 0, which in turn leads to q = 0, or to the study of rational points on the hyperelliptic quartic curve…”

“…Tying now all these results together we get the statement of the theorem. For all (m, k) except (4, 2) and (5,4), the variety that parameterizes such examples is a non-hyperelliptic curve of genus 4, and so other than searching for small points (resulting in the above examples) we have not carried investigation further here.…”

“…Now the Jacobian of C has rank 1, and Magma's Chabauty routines determine that the only finite rational points on C are (X, ±Y ) = (0, 0), (1,4). This in turn gives (p, q) = (0, 0), (1, 1), ( 1 2 , 3 8 ) as the complete set of finite rational points on (2), leading only to a = 0.…”