Acta Mathematica Hungarica

, Volume 145, Issue 2, pp 320–349 | Cite as

Some observations concerning reducibility of quadrinomials

  • A. Bremner
  • M. UlasEmail author


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 \({\geqq 3}\).

Key words and phrases

quadrinomial factorization reducibility curve of genus 2 

Mathematics Subject Classification

13P05 11C08 11G30 


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Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2015

Authors and Affiliations

  1. 1.School of Mathematical and Statistical SciencesArizona State UniversityTempeUSA
  2. 2.Faculty of Mathematics and Computer Science, Institute of MathematicsKrakówPoland

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