On the reducibility of some special quadrinomials

TL;DR

This paper investigates the reducibility of specific quadrinomials with degrees 4 and 5, linking their factorization properties to the existence of rational points on associated elliptic and hyperelliptic curves.

Contribution

It provides new insights into the reducibility conditions of certain quadrinomials and their connection to rational points on elliptic and hyperelliptic curves.

Findings

Characterization of reducibility conditions for quadrinomials with degree 4 and 5.

Identification of links between polynomial reducibility and rational points on elliptic/hyperelliptic curves.

New examples of reducible quadrinomials related to rational solutions.

Abstract

Let $n>m>k$ be positive integers and let $a,b,c$ be nonzero rational numbers. We consider the reducibility of some special quadrinomials $x^n+ax^m+bx^k+c$ with $n=4$ and 5, which related to the study of rational points on certain elliptic curves or hyperelliptic curves.