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

Citations:

Zbl 1243.11115

References:

[1]	Hardy, An introduction to the theory of numbers (1979) · Zbl 0423.10001
[2]	DOI: 10.1016/j.ffa.2010.06.005 · Zbl 1222.11143
[3]	DOI: 10.1023/A:1000130114331 · Zbl 0886.11016
[4]	DOI: 10.1007/BF01974110 · Zbl 0552.10009
[5]	Blake, Application of finite fields (1993)
[6]	Stichtenoth, Algebraic function fields and codes (1993) · Zbl 1155.14022
[7]	DOI: 10.1017/S0305004100044418
[8]	Ayad, Acta Arith. 93 pp 87– (2000)
[9]	DOI: 10.1112/plms/s3-51.3.385 · Zbl 0622.12011
[10]	DOI: 10.4064/aa119-1-4 · Zbl 1088.11078
[11]	DOI: 10.1112/jlms/jdn034 · Zbl 1193.37144
[12]	Jones, Compositio Math. 43 pp 1108– (2007) · Zbl 1166.11040
[13]	Iwaniec, Analytic number theory (2004)
[14]	DOI: 10.1090/S0002-9939-10-10404-3 · Zbl 1268.11155