On stable quadratic polynomials. (English) Zbl 1241.11027

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 \(\alpha_d=\log 2/2\log(4d)\).
It was shown by R. Jones and 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.


11C08 Polynomials in number theory
11T06 Polynomials over finite fields
37P05 Arithmetic and non-Archimedean dynamical systems involving polynomial and rational maps


Zbl 1243.11115
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