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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2501.07409 |
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Table of Contents:
- Let $K$ be a field and $ϕ(z)\in K[z]$ be a polynomial. Define $Φ(z) := \frac{1}{ϕ(z)} \in K(z).$ For $n \in\mathbb{N}^* $, let the $n$-th iterate of $Φ(z)$ be defined as $Φ^{(n)}(z) = \underbrace{Φ\circ Φ\circ \cdots \circ Φ}_{n \text{ times}}(z).$ We express the \(Φ^{(n)}(z)\) in its reduced form as \( Φ^{(n)}(z) = \frac{f_{n,ϕ}(z)}{g_{n,ϕ}(z)}, \) where \(f_{n,ϕ}(z)\) and \(g_{n,ϕ}(z)\) are coprime polynomials in \(K[z]\). A polynomial $ϕ(z) \in K[z]$ is called inversely stable over $K$ if every $g_{n,ϕ}(z)$ in the sequence $\{g_{n,ϕ}(z)\}_{n=1}^\infty$ is irreducible in $K[z]$. This paper investigates the inverse stability of the binomials $ϕ(z) = z^d + c$ over $K$.