On the inverse stability of $z^n+c.$
Yang Gao Qingzhong Ji
TL;DR
This work establishes criteria for the inverse stability of binomial polynomials $\\phi(z)=z^d+c$ across number fields, function fields, and finite fields by studying the irreducibility of the numerators $f_{n,\\phi}$ and denominators $g_{n,\\phi}$ in iterates of $\\Phi(z)=1/\\phi(z)$. The authors prove that irreducibility of $\\phi$ together with explicit local obstructions on $c$ (e.g., $c\notin uR^p$ for primes $p|d$) implies inverse stability, and they derive several concrete consequences: over $\\mathbb{Q}$ (odd $d$ or even $d$ with $c$ non-square), over function fields of characteristic $0$, and over finite fields under rad$(d)$-free type conditions; they also provide a recurrence-based criterion for rad$(d)$-freeness and prove a density result for inversely stable binomials over certain finite fields. The results yield connections to eventual stability in arithmetic dynamics, offer a method to generate infinite families of irreducible polynomials over finite fields, and have implications for Arboreal Galois representations.
Abstract
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 $K$.