×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI
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 · doi:10.1016/j.ffa.2010.06.005
[3] DOI: 10.1023/A:1000130114331 · Zbl 0886.11016 · doi:10.1023/A:1000130114331
[4] DOI: 10.1007/BF01974110 · Zbl 0552.10009 · doi:10.1007/BF01974110
[5] Blake, Application of finite fields (1993) · doi:10.1007/978-1-4757-2226-0
[6] Stichtenoth, Algebraic function fields and codes (1993) · Zbl 1155.14022
[7] DOI: 10.1017/S0305004100044418 · doi:10.1017/S0305004100044418
[8] Ayad, Acta Arith. 93 pp 87– (2000)
[9] DOI: 10.1112/plms/s3-51.3.385 · Zbl 0622.12011 · doi:10.1112/plms/s3-51.3.385
[10] DOI: 10.4064/aa119-1-4 · Zbl 1088.11078 · doi:10.4064/aa119-1-4
[11] DOI: 10.1112/jlms/jdn034 · Zbl 1193.37144 · doi:10.1112/jlms/jdn034
[12] Jones, Compositio Math. 43 pp 1108– (2007) · Zbl 1166.11040 · doi:10.1112/S0010437X07002667
[13] Iwaniec, Analytic number theory (2004) · doi:10.1090/coll/053
[14] DOI: 10.1090/S0002-9939-10-10404-3 · Zbl 1268.11155 · doi:10.1090/S0002-9939-10-10404-3
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.