On the inverse stability of $z^n+c.$

Abstract

Let $K$ be a field and $\phi(z)\in K[z]$ be a polynomial. Define $\Phi(z) := \frac{1}{\phi(z)} \in K(z).$ For $n \in\mathbb{N}^* $, let the $n$-th iterate of $\Phi(z)$ be defined as $\Phi^{(n)}(z) = \underbrace{\Phi \circ \Phi \circ \cdots \circ \Phi}_{n \text{ times}}(z).$ We express the \(\Phi^{(n)}(z)\) in its reduced form as \( \Phi^{(n)}(z) = \frac{f_{n,\phi}(z)}{g_{n,\phi}(z)}, \) where \(f_{n,\phi}(z)\) and \(g_{n,\phi}(z)\) are coprime polynomials in \(K[z]\). A polynomial $\phi(z) \in K[z]$ is called inversely stable over $K$ if every $g_{n,\phi}(z)$ in the sequence $\{g_{n,\phi}(z)\}_{n=1}^\infty$ is irreducible in $K[z]$. This paper investigates the inverse stability of the binomials $\phi(z) = z^d + c$ over $K$.