On stable quadratic polynomials
type: Item
instance of: scholarly article
zbMATH Open document ID: 1241.11027
DOI: 10.1017/S001708951200002X
author: Omran Ahmadi
author: Florian Luca
author: Alina Ostafe
author: Igor E. Shparlinski
published in: Glasgow Mathematical Journal
publication date: 2012-05-06T00:00:00Z
Mathematics Subject Classification ID: 11C08
Mathematics Subject Classification ID: 11T06
Mathematics Subject Classification ID: 37P05
zbMATH DE Number: 6031018
zbMATH Keywords: stable polynomials
zbMATH Keywords: irreducibility of polynomials
zbMATH Keywords: iterated polynomials
MaRDI profile type: Publication
cites work: Q3856819
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cites work: The Galois Theory of Iterates and Composites of Polynomials
cites work: Stabilité des polynômes
cites work: The density of prime divisors in the arithmetic dynamics of quadratic polynomials
cites work: Iterated Galois towers, their associated martingales, and the $p$-adic Mandelbrot set
cites work: On the length of critical orbits of stable quadratic polynomials
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title: On stable quadratic polynomials
review text: 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.
reviewed by: Władysław Narkiewicz
description: scientific article; zbMATH DE number 6031018