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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
Software:
Magma; LMFDB
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References:
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