Some observations concerning reducibility of quadrinomials

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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<sup>n</sup>+x<sup>m</sup>+x<sup>k</sup>+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.

Original languageEnglish (US)
Pages (from-to)320-349
Number of pages30
JournalActa Mathematica Hungarica
Volume145
Issue number2
DOIs
StatePublished - Apr 1 2015

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Keywords

  • curve of genus 2
  • factorization
  • quadrinomial
  • reducibility

ASJC Scopus subject areas

  • Mathematics(all)

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