On Monogeneity of reciprocal polynomials
Rupam Barman, Anuj Narode, Vinay Wagh
TL;DR
The paper investigates monogeneity for reciprocal polynomials, focusing on even-degree cases and exploiting a discriminant relation that links the polynomial and its associated Chebyshev-type transform $g(x)$ via $f(x)=x^{n}g(x+x^{-1})$. By combining discriminant formulas with results on irreducibility and index theory, it provides a general criterion: if $f(-1)f(1)$ is squarefree and $g$ is monogenic, then $f$ is monogenic, and it derives corollaries for power-compositional polynomials. The authors construct infinite families of monogenic polynomials of degrees $10$ and $5$ (under the $abc$-conjecture for number fields) and prove a discriminant factorization for certain cyclotomic-modified families, extending Jones's conjectures in part. They also prove a lower bound $\, ext{for the number of monogenic sextic reciprocal polynomials}$, showing that a positive proportion of such polynomials are monogenic as $|a|$ grows. Collectively, the work advances explicit criteria and constructions for monogeneity in reciprocal polynomials and connects discriminants, indices, and Galois-theoretic considerations to quantify monogenic instances.
Abstract
Let $\mathbb{Z}_K$ denote the ring of integers of the number field $K = \mathbb{Q}(θ)$, where $θ$ is a root of the monic irreducible polynomial $f(x) \in \mathbb{Z}[x]$. We say that $f(x)$ is monogenic if $\mathbb{Z}_K = \mathbb{Z}[θ]$. 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.