On the reducibility type of trinomials

TL;DR

This paper investigates the reducibility types of trinomials over the rationals by analyzing the associated algebraic curves of genus 0, 1, and 2, aiming to classify all possible factorizations based on rational points.

Contribution

It characterizes reducibility types of trinomials through algebraic curves of low genus, providing a framework to describe all such types when the genus is 0, 1, or 2.

Findings

Classification of reducibility types for low genus cases

Description of rational points on algebraic curves related to trinomials

Methodology for determining all reducibility types in these cases

Abstract

Say a trinomial $x^n+A x^m+B \in \Q[x]$ has reducibility type $(n_1,n_2,...,n_k)$ if there exists a factorization of the trinomial into irreducible polynomials in $\Q[x]$ of degrees $n_1$, $n_2$,...,$n_k$, ordered so that $n_1 \leq n_2 \leq ... \leq n_k$. Specifying the reducibility type of a monic polynomial of fixed degree is equivalent to specifying rational points on an algebraic curve. When the genus of this curve is 0 or 1, there is reasonable hope that all its rational points may be described; and techniques are available that may also find all points when the genus is 2. Thus all corresponding reducibility types may be described. These low genus instances are the ones studied in this paper.