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    This article has been cited by the following publications. This list is generated based on data provided by CrossRef.

    Goksel, Vefa Xia, Shixiang and Boston, Nigel 2015. A Refined Conjecture for Factorizations of Iterates of Quadratic Polynomials over Finite Fields. Experimental Mathematics, Vol. 24, Issue. 3, p. 304.

    2013. Handbook of Finite Fields.

    Jones, Rafe 2012. An iterative construction of irreducible polynomials reducible modulo every prime. Journal of Algebra, Vol. 369, p. 114.



  • DOI:
  • Published online: 29 March 2012

We recall that a polynomial f(X) ∈ K[X] over a field K is called stable if all its iterates are irreducible over K. We show that almost all monic quadratic polynomials f(X) ∈ ℤ[X] are stable over ℚ. We also show that the presence of squares in so-called critical orbits of a quadratic polynomial f(X) ∈ ℤ[X] can be detected by a finite algorithm; this property is closely related to the stability of f(X). We also prove there are no stable quadratic polynomials over finite fields of characteristic 2 but they exist over some infinite fields of characteristic 2.

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1.N. Ali , Stabilité des polynômes, Acta Arith. 119 (2005), 5363.

4.I. F. Blake , X. H. Gao , A. J. Menezes , R. C. Mullin , S. A. Vanstone and T. Yaghoobian , Application of finite fields (Kluwer, 1993).

5.B. Brindza , On S-integral solutions of the equation ym = f(x), Acta Math. Hungar. 44 (1984), 133139.

7.D. Gomez and A. P. Nicolás , An estimate on the number of stable quadratic polynomials, Finite Fields Appl. 16 (2010), 329333.

10.H. Iwaniec and E. Kowalski , Analytic number theory (Amer. Math. Soc. Providence, RI, 2004).

12.R. Jones , The density of prime divisors in the arithmetic dynamics of quadratic polynomials, J. London Math. Soc. 78 (2008), 523544.

15.A. Ostafe and I. E. Shparlinski , On the length of critical orbits of stable quadratic polynomials, Proc. Amer. Math. Soc. 138 (2010), 26532656.

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Glasgow Mathematical Journal
  • ISSN: 0017-0895
  • EISSN: 1469-509X
  • URL: /core/journals/glasgow-mathematical-journal
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