THE MONOGENICITY OF POWER-COMPOSITIONAL CHARACTERISTIC POLYNOMIALS

Source abstract

Let $f(x) \in \mathbb{Z}[x]$ be monic with $\mathrm{deg}(f) = N \geq 2$. Suppose that $f(x)$ is monogenic, and that $f(x)$ is the characteristic polynomial of the $N$th order linear recurrence sequence $\Upsilon_f := (U_n)_{n \geq 0}$ with initial conditions $$U_0 = U_1 = \cdots = U_{N−2} = 0 \mathrm{\hspace {1 pc}and\hspace {1 pc}} U_{N−1} = 1.$$ Let $\pi (m)$ denote the length of the period of $\Upsilon_f$ modulo the integer $m \geq 2$, where $\mathrm{gcd}(m, f(0)) = 1$. 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 $\pi (p^2) \neq \pi(p)$, which provides a new and simple test for the monogenicity of $f(x^p)$. We also present some infinite families of such polynomials. This article extends previous work of the author.