On stable quadratic polynomials (Q2882500)

Let \(K\) be a field. A polynomial \(f\in K[X]\) is called stable if all its iterates \(f^{(1)}=f, f^{(2)}=f(f),\dots, f^{(n)},\dots\) are irreducible over \(K\). The main result of the paper states that almost all irreducible quadratic polynomials in \(\mathbb Z[X]\) are stable (Theorem 1), but there are no stable quadratic polynomials over a finite field of characteristic \(2\) (Corollary 11). In the case of finite fields \(\mathbb F_q\) of odd characteristic it is shown (Theorem 8) that if \(F(X)=g(aX^2+bX+c)\) is stable and \(\deg g=d\), then the orbit of \(-b/2a\) under \(F\) has \(O(q^{1-\alpha_d})\) elements with \(\alpha_d=\log 2/2\log(4d)\).NEWLINENEWLINEIt was shown by \textit{R. Jones} and \textit{N. Boston} [Proc. Am. Math. Soc., 140, No. 6, 1849--1863 (2012; Zbl 1243.11115)] that if \(f(X)=aX^2+bX+c\) (with \(a\neq 0\)), \(\gamma=-b/2a\) and the sequence \(-f(\gamma),f^{(2)}(\gamma),\dots,f^{(n)}(\gamma),\dots\) contains no squares, then \(f\) is stable. The authors present (Theorem 5) an effective algorithm based on Baker's method to test whether the assumption of this assertion is satisfied.