On stable quadratic polynomials
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.
scientific article; zbMATH DE number 6031018
scientific article