Stability of Certain Higher Degree Polynomials

Abstract

One of the interesting problems in arithmetic dynamics is to study the stability of polynomials over a field. In this paper, we study the stability of $f(z)=z^d+\frac{1}{c}$ for $d\ge 2$, $c\in \mathbb{Z}\setminus \{0\}$. We show that for infinite families of $d\ge 3$, whenever $f(z)$ is irreducible, all its iterates are irreducible, that is, $f(z)$ is stable. For $c\equiv 1(\mathrm{mod}4)$, we show that all the iterates of $z^2+\frac{1}{c}$ are irreducible. Also we show that for $d=3$, if $f(z)$ is reducible, then the number of irreducible factors of each iterate of $f(z)$ is exactly $2$ for $|c|\le 10^{12}$.