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 $K$.
Categories: math.DS math.NT