Albanian J. Math. 15 (2), 85-98, (2021) DOI: 10.51286/albjm/1635847634
KEYWORDS: stable polynomial, monogenic polynomial, irreducible polynomial, 11R06, 12F05, 11R09

A monic polynomial $f\left(x\right)\in \mathrm{\mathbb{Z}}\left[x\right]$ is called *stable* if ${f}^{n}\left(x\right)$ is irreducible over $\mathrm{\mathbb{Q}}$ for all $n\ge 1$, where ${f}^{n}\left(x\right)$ denotes the $n$th iterate of $f\left(x\right)$. Regardless of whether $f\left(x\right)$ is irreducible over $\mathrm{\mathbb{Q}}$, if there exists some monic $g\left(x\right)\in \mathrm{\mathbb{Z}}\left[x\right]$ such that $g\left({f}^{n}\right(x\left)\right)$ is irreducible over $\mathrm{\mathbb{Q}}$ for all $n\ge 1$, we say that $f\left(x\right)$ is $g$*-stable*. Many authors have studied such polynomials since Odoni first introduced this concept of stability in 1985. We extend these concepts here by adding the additional restriction of monogeneity. A monic polynomial $f\left(x\right)\in \mathrm{\mathbb{Z}}\left[x\right]$ is defined to be *monogenic* if $f\left(x\right)$ is irreducible over $\mathrm{\mathbb{Q}}$ and $\left\{1,\theta ,{\theta}^{2},\dots ,{\theta}^{\mathrm{deg}\left(f\right)-1}\right\}$ is a basis for the ring of integers of $\mathrm{\mathbb{Q}}\left(\theta \right)$, where $f\left(\theta \right)=0$. We say that $f\left(x\right)$ is $g$*-monogenically stable*, if $g\left({f}^{n}\right(x\left)\right)$ is monogenic for all $n\ge 1$, for some monic $g\left(x\right)\in \mathrm{\mathbb{Z}}\left[x\right]$. When $g\left(x\right)=x$, we simply say that $f\left(x\right)$ is *monogenically stable*. In this article, we provide methods for constructing $g$-monogenically stable polynomials $f\left(x\right)$, for various polynomials $f\left(x\right)$ and $g\left(x\right)$.