On Monogeneity of reciprocal polynomials

TL;DR

This paper investigates conditions under which even degree reciprocal polynomials generate monogenic number fields, providing partial proofs of a recent conjecture and establishing bounds on specific sextic cases.

Contribution

It offers new sufficient conditions for monogeneity of even degree reciprocal polynomials and advances understanding of Jones's 2021 conjecture.

Findings

Derived sufficient conditions for monogeneity of even degree reciprocal polynomials.

Partially proved Jones's conjecture from 2021.

Established a lower bound on the count of certain sextic monogenic reciprocal polynomials.

Abstract

Let $\mathbb{Z}_K$ denote the ring of integers of the number field $K = \mathbb{Q}(\theta)$, where $\theta$ is a root of the monic irreducible polynomial $f(x) \in \mathbb{Z}[x]$. We say that $f(x)$ is monogenic if $\mathbb{Z}_K = \mathbb{Z}[\theta]$. A polynomial $f(x) \in \mathbb{Z}[x]$ is called reciprocal if $f(x) = x^{\operatorname{deg}(f)} f(1/x)$. In this article, we derive sufficient conditions for the monogeneity of even degree reciprocal polynomials. By employing properties of the discriminant of reciprocal polynomials, we partially prove a conjecture proposed by Jones in $2021$. Furthermore, we establish a lower bound on the number of certain sextic monogenic reciprocal polynomials.