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Eventually stable quadratic polynomials over $$\mathbb{Q}$$. (English) Zbl 1446.37100
Let $$K$$ be a field, $$\alpha \in K$$, $$f \in K[x]$$. A pair $$(f, \alpha)$$ is called eventually stable over $$K$$ if there exists a constant $$C(f;\alpha)$$ such that the number of irreducible factors over $$K$$ of $$f^n(x)-\alpha$$, where $$f^n$$ stands for the $$n$$-th iterate of $$f$$, is at most $$C(f;\alpha)$$ for all $$n\geq 1$$. Also, $$f$$ is eventually stable over $$K$$ if $$(f; 0)$$ is eventually stable.
The authors prove that the polynomial $$f_c(x) = x^2 + 1/c$$ is eventually stable over $$\mathbb Q$$ for $$c \in {\mathbb Z} \setminus \{0,-1\}$$ satisfying $$|c| \leq 10^9$$, and that $$C(f_c,0) \leq 4$$. They also describe many series of $$c$$ when the $$n$$-th iterate of $$f_c$$ is irreducible over $$\mathbb Q$$ for all $$n \geq 1$$.
##### MSC:
 37P15 Dynamical systems over global ground fields 11R09 Polynomials (irreducibility, etc.) 37P05 Arithmetic and non-Archimedean dynamical systems involving polynomial and rational maps 12E05 Polynomials in general fields (irreducibility, etc.) 11R32 Galois theory 11R45 Density theorems
LMFDB; Magma
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