Monogenic trinomials with non-squarefree discriminant

TL;DR

This paper identifies new infinite families of monogenic trinomials with non-squarefree discriminant, many having small Galois groups, and provides asymptotic counts for certain parameter ranges.

Contribution

It introduces novel infinite families of monogenic trinomials with non-squarefree discriminant, expanding beyond previously studied specific forms.

Findings

Identified new infinite families of monogenic trinomials with non-squarefree discriminant.

Many of these families have small Galois groups.

Provided asymptotic estimates for the number of such trinomials with bounded parameters.

Abstract

For each integer $n\ge 2$, we identify new infinite families of monogenic trinomials $f(x)=x^n+Ax^m+B$ with non-squarefree discriminant, many of which have small Galois group. These families are thus different from many previous examinations of specific trinomial forms in the literature. Moreover, in certain situations when $A=B\ge 2$ with fixed $n$ and $m$, we produce asymptotics on the number of such trinomials with $A\le X$.