2021 MONOGENICALLY STABLE POLYNOMIALS
Lenny Jones
Albanian J. Math. 15(2): 85-98 (2021). DOI: 10.51286/albjm/1635847634

## Abstract

A monic polynomial $f(x)\in\mathbb{Z}\lbrack x\rbrack$ is called stable if $f^n(x)$ is irreducible over $\mathbb{Q}$ for all $n\geq1$, where $f^n(x)$ denotes the $n$th iterate of $f(x)$. Regardless of whether $f(x)$ is irreducible over $\mathbb{Q}$, if there exists some monic $g(x)\in\mathbb{Z}\lbrack x\rbrack$ such that $g(f^n(x))$ is irreducible over $\mathbb{Q}$ for all $n\geq1$, we say that $f(x)$ 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 $f(x)\in\mathbb{Z}\lbrack x\rbrack$ is defined to be monogenic if $f(x)$ is irreducible over $\mathbb{Q}$ and $\left\{1,\theta,\theta^2,\dots,\theta^{\mathrm{def}\left(f\right)-1}\right\}$ is a basis for the ring of integers of $\mathbb{Q}\left(\theta\right)$, where $f(\theta)=0$. We say that $f(x)$ is $g$-monogenically stable, if $g(f^n(x))$ is monogenic for all $n\geq1$, for some monic $g(x)\in\mathbb{Z}\lbrack x\rbrack$. When $g(x)=x$, we simply say that $f(x)$ is monogenically stable. In this article, we provide methods for constructing $g$-monogenically stable polynomials $f(x)$, for various polynomials $f(x)$ and $g(x)$.

## Citation

Lenny Jones. "MONOGENICALLY STABLE POLYNOMIALS." Albanian J. Math. 15 (2) 85 - 98, 2021. https://doi.org/10.51286/albjm/1635847634

## Information

Published: 2021
First available in Project Euclid: 11 July 2023

MathSciNet: MR4346263
zbMATH: 1490.11107
Digital Object Identifier: 10.51286/albjm/1635847634

Subjects:
Primary: 11R06 , 11R09 , 12F05

Keywords: irreducible polynomial , monogenic polynomial , stable polynomial