an:07213814
Zbl 1446.37100
Demark, David; Hindes, Wade; Jones, Rafe; Misplon, Moses; Stoll, Michael; Stoneman, Michael
Eventually stable quadratic polynomials over \(\mathbb{Q}\)
EN
New York J. Math. 26, 526-561 (2020).
00445013
2020
j
37P15 11R09 37P05 12E05 11R32 11R45
iterated polynomials; irreducible polynomials; rational points; hyperelliptic curves; arboreal Galois representation
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 \(n \geq 1\).
Art??ras Dubickas (Vilnius)
Zbl 1395.11128