On stable quadratic polynomials. (English) Zbl 1241.11027

Ahmadi, Omran; Luca, Florian; Ostafe, Alina; Shparlinski, Igor E.

Glasg. Math. J. 54, No. 2, 359-369 (2012).

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.

Reviewer: Władysław Narkiewicz (Wrocław)

MSC:

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

Keywords:

stable polynomials; irreducibility of polynomials; iterated polynomials

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