The Monogenicity of Power-Compositional Characteristic Polynomials

Abstract

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 ${{\rm Y}}_f:=(U_n{)}_{n\ge 0}$ 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 $\pi (p^2)\ne \pi (p)$, where $\pi (m)$ denotes the period of ${{\rm Y}}_f$ 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.