Stable quadratic polynomials over $\mathbb{Q}(i)$

Abstract

We study iterates of a quadratic $f= x^2+1/c\in K[x]$. If the number of factors of $f^n:=f\circ f \circ ... \circ f$ 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 $\mathbb{Q}$. The conjecture "if $f^2$ is irreducible, then $f^n$ is irreducible for all $n$" extends to $\mathbb{Q}(i)$, but due to the lack of a linear ordering on $\mathbb{Q}(i)$, an auxiliary function is involved in a specific $n$ to check. The elusive case of $c\equiv 2 \bmod 4$ (as a $\mathbb{Z}$ equivalence class) is shown to be "stable" over $\mathbb{Q}(i)$, offering more evidence for \cite[Conjecture 1]{evstb}. Stability for $c\equiv 1\bmod 2$ (as a $\mathbb{Z}[i]$ equivalence class) is not as fully handled as over $\mathbb{Z}$, however.