### 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 language | English (US) |
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Pages (from-to) | 320-349 |

Number of pages | 30 |

Journal | Acta Mathematica Hungarica |

Volume | 145 |

Issue number | 2 |

DOIs | |

State | Published - Apr 1 2015 |

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### Keywords

- curve of genus 2
- factorization
- quadrinomial
- reducibility

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Acta Mathematica Hungarica*,

*145*(2), 320-349. https://doi.org/10.1007/s10474-015-0478-9