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 |
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Keywords
- curve of genus 2
- factorization
- quadrinomial
- reducibility
ASJC Scopus subject areas
- Mathematics(all)
Cite this
Some observations concerning reducibility of quadrinomials. / Bremner, Andrew; Ulas, M.
In: Acta Mathematica Hungarica, Vol. 145, No. 2, 01.04.2015, p. 320-349.Research output: Contribution to journal › Article
}
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 -