On Monogeneity of reciprocal polynomials

Abstract

Let ${\mathbb{Z}}_K$ denote the ring of integers of the number field $K=\mathbb{Q}(\hat{I}¸)$, where $\hat{I}¸$ is a root of the monic irreducible polynomial $f(x)\in \mathbb{Z}[x]$. We say that $f(x)$ is monogenic if ${\mathbb{Z}}_K=\mathbb{Z}[\hat{I}¸]$. A polynomial $f(x)\in \mathbb{Z}[x]$ is called reciprocal if $f(x)=x^{\mathrm{deg}(f)}f(1/x)$. 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.

To apper in The Ramanujan Journal