The Monogenicity of Power-Compositional Characteristic Polynomials
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866910464605159424 |
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| author | Jones, Lenny |
| author_facet | Jones, Lenny |
| contents | Let $f(x)\in {\mathbb Z}[x]$ be monic of degree $N\ge 2$. Suppose that $f(x)$ is monogenic, and that $f(x)$ is the characteristic polynomial of the $N$th order linear recurrence sequence $Υ_f:=(U_n)_{n\ge 0}$ with initial conditions \[U_0=U_1=\cdots =U_{N-2}=0 \quad \mbox{and} \quad U_{N-1}=1.\] Let $p$ be a prime such that $f(x)$ is irreducible over ${\mathbb F}_p$ and $f(x^p)$ is irreducible over ${\mathbb Q}$. We prove that $f(x^p)$ is monogenic if and only if $π(p^2)\ne π(p)$, where $π(m)$ denotes the period of $Υ_f$ modulo $m$. These results extend previous work of the author, and provide a new and simple test for the monogenicity of $f(x^p)$. We also provide some infinite families of such polynomials. This article extends previous work of the author. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_08875 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The Monogenicity of Power-Compositional Characteristic Polynomials Jones, Lenny Number Theory Let $f(x)\in {\mathbb Z}[x]$ be monic of degree $N\ge 2$. Suppose that $f(x)$ is monogenic, and that $f(x)$ is the characteristic polynomial of the $N$th order linear recurrence sequence $Υ_f:=(U_n)_{n\ge 0}$ with initial conditions \[U_0=U_1=\cdots =U_{N-2}=0 \quad \mbox{and} \quad U_{N-1}=1.\] Let $p$ be a prime such that $f(x)$ is irreducible over ${\mathbb F}_p$ and $f(x^p)$ is irreducible over ${\mathbb Q}$. We prove that $f(x^p)$ is monogenic if and only if $π(p^2)\ne π(p)$, where $π(m)$ denotes the period of $Υ_f$ modulo $m$. These results extend previous work of the author, and provide a new and simple test for the monogenicity of $f(x^p)$. We also provide some infinite families of such polynomials. This article extends previous work of the author. |
| title | The Monogenicity of Power-Compositional Characteristic Polynomials |
| topic | Number Theory |
| url | https://arxiv.org/abs/2311.08875 |