### 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) |
---|---|

Pages (from-to) | 320-349 |

Number of pages | 30 |

Journal | Acta Mathematica Hungarica |

Volume | 145 |

Issue number | 2 |

DOIs | |

State | Published - Apr 1 2015 |

### Fingerprint

### 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

**Some observations concerning reducibility of quadrinomials.** / Bremner, Andrew; Ulas, M.

Research output: Contribution to journal › Article

*Acta Mathematica Hungarica*, vol. 145, no. 2, pp. 320-349. https://doi.org/10.1007/s10474-015-0478-9

}

TY - JOUR

T1 - Some observations concerning reducibility of quadrinomials

AU - Bremner, Andrew

AU - Ulas, M.

PY - 2015/4/1

Y1 - 2015/4/1

N2 - 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)=xn+xm+xk+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.

AB - 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)=xn+xm+xk+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.

KW - curve of genus 2

KW - factorization

KW - quadrinomial

KW - reducibility

UR - http://www.scopus.com/inward/record.url?scp=84939969732&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84939969732&partnerID=8YFLogxK

U2 - 10.1007/s10474-015-0478-9

DO - 10.1007/s10474-015-0478-9

M3 - Article

AN - SCOPUS:84939969732

VL - 145

SP - 320

EP - 349

JO - Acta Mathematica Hungarica

JF - Acta Mathematica Hungarica

SN - 0236-5294

IS - 2

ER -