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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.

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