TY - JOUR

T1 - Some observations concerning reducibility of quadrinomials

AU - Bremner, Andrew

AU - Ulas, M.

N1 - Publisher Copyright:
© 2015, Akadémiai Kiadó, Budapest, Hungary.

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

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U2 - 10.1007/s10474-015-0478-9

DO - 10.1007/s10474-015-0478-9

M3 - Article

AN - SCOPUS:84939969732

SN - 0236-5294

VL - 145

SP - 320

EP - 349

JO - Acta Mathematica Hungarica

JF - Acta Mathematica Hungarica

IS - 2

ER -