Stable Quadratic Polynomials over $\mathbb{Q}(i)$

Abstract

Let $f$ be a polynomial or a rational function over a field $K$. A basic question is, if $f$ is a polynomial, are its iterates irreducible or not? We wish to know what can happen when considering iterates of a quadratic $f= x^2+r\in K[x]$. If the number of factors of $f^n$ is bounded by a constant independent of $n$, then $f$ is said to be \emph{eventually stable}. This paper is an extension to $\mathbb{Q}(i)$ of the paper \cite{evstb}, which considered $f$ over $\mathbb{Q}$. Showing stability for $c\equiv 1 \mod 2$ (as a $\mathbb{Z}[i]$ equivalence class) is not as fully handled as over $\mathbb{Z}$, however, the elusive case of $c\equiv 2 \mod 4$ (as a $\mathbb{Z}$ equivalence class) is shown to be stable over $\mathbb{Z}[i]$, offering more evidence for \cite[Conjecture 1]{evstb}. The conjecture "if $f^2$ is irreducible, then $f^n$ is irreducible for all $n$" extends to $\mathbb{Q}(i)$, and due to the lack of a linear ordering on $\mathbb{Q}(i)$, a new function is involved in a specific $n$ to check.