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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2501.07409 |
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| _version_ | 1866929711412674560 |
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| author | Ji, Yang Gao Qingzhong |
| author_facet | Ji, Yang Gao Qingzhong |
| 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 $K$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_07409 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the inverse stability of $z^n+c.$ Ji, Yang Gao Qingzhong Dynamical Systems Number Theory 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 $K$. |
| title | On the inverse stability of $z^n+c.$ |
| topic | Dynamical Systems Number Theory |
| url | https://arxiv.org/abs/2501.07409 |