On Monogeneity of reciprocal polynomials
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866917233381343232 |
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| author | Barman, Rupam Narode, Anuj Wagh, Vinay |
| author_facet | Barman, Rupam Narode, Anuj Wagh, Vinay |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_22453 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On Monogeneity of reciprocal polynomials Barman, Rupam Narode, Anuj Wagh, Vinay Number Theory 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. |
| title | On Monogeneity of reciprocal polynomials |
| topic | Number Theory |
| url | https://arxiv.org/abs/2601.22453 |