Hostname: page-component-74656f9884-8r8nv Total loading time: 0 Render date: 2026-11-09T09:19:07.303Z Has data issue: false hasContentIssue false
  • English
  • Français

ON STABLE QUADRATIC POLYNOMIALS

Published online by Cambridge University Press:  29 March 2012

OMRAN AHMADI ,
FLORIAN LUCA ,
ALINA OSTAFE  and
IGOR E. SHPARLINSKI
OMRAN AHMADI
Affiliation:
Claude Shannon Institute, University College Dublin, Dublin 4, Ireland e-mail: omran.ahmadi@ucd.ie
FLORIAN LUCA
Affiliation:
Instituto de Matemáticas, Universidad Nacional Autonoma de México, C.P. 58089, Morelia, Michoacán, Mexico e-mail: fluca@matmor.unam.mx
ALINA OSTAFE
Affiliation:
Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190 CH-8057, Zürich, Switzerland e-mail: alina.ostafe@math.uzh.ch
IGOR E. SHPARLINSKI
Affiliation:
Department of Computing, Macquarie University, Sydney, NSW 2109, Australia e-mail: igor.shparlinski@mq.edu.au
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

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.

Keywords

11C0811T0637P05

Information

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 54 , Issue 2 , May 2012 , pp. 359 - 369
Copyright
Copyright © Glasgow Mathematical Journal Trust 2012

References

REFERENCES

1.Ali, N., Stabilité des polynômes, Acta Arith. 119 (2005), 53–63.CrossRefGoogle Scholar
2.Ayad, M. and McQuillan, D. L., Irreducibility of the iterates of a quadratic polynomial over a field, Acta Arith. 93 (2000), 87–97; Corrigendum: Acta Arith. 99 (2001), 97.CrossRefGoogle Scholar
3.Baker, A., Bounds for the solutions of the hyperelliptic equation, Proc. Cambridge Philos. Soc. 65 (1969), 439–444.CrossRefGoogle Scholar
4.Blake, I. F., Gao, X. H., Menezes, A. J., Mullin, R. C., Vanstone, S. A. and Yaghoobian, T., Application of finite fields (Kluwer, 1993).CrossRefGoogle Scholar
5.Brindza, B., On S-integral solutions of the equation ym = f(x), Acta Math. Hungar. 44 (1984), 133–139.CrossRefGoogle Scholar
6.Bugeaud, Y., Bounds for the solutions of superelliptic equations, Compositio Math. 107 (1997), 187–219.CrossRefGoogle Scholar
7.Gomez, D. and Nicolás, A. P., An estimate on the number of stable quadratic polynomials, Finite Fields Appl. 16 (2010), 329–333.CrossRefGoogle Scholar
8.Gomez-Perez, D., Nicolás, A. P., Ostafe, A. and Sadornil, D., Stable polynomials over finite fields, Preprint (2011).Google Scholar
9.Hardy, G. H. and Wright, E. M., An introduction to the theory of numbers (Oxford University Press, Oxford, 1979).Google Scholar
10.Iwaniec, H. and Kowalski, E., Analytic number theory (Amer. Math. Soc. Providence, RI, 2004).Google Scholar
11.Jones, R., Iterated Galois towers, associated martingales, and the p-adic Mandelbrot set, Compositio Math. 43 (2007), 1108–1126.CrossRefGoogle Scholar
12.Jones, R., The density of prime divisors in the arithmetic dynamics of quadratic polynomials, J. London Math. Soc. 78 (2008), 523–544.CrossRefGoogle Scholar
13.Jones, R. and Boston, N., Settled polynomials over finite fields, Proc. Amer. Math. Soc. (to appear).Google Scholar
14.Odoni, R. W. K., The Galois theory of iterates and composites of polynomials, Proc. London Math. Soc. 51 (1985), 385–414.CrossRefGoogle Scholar
15.Ostafe, A. and Shparlinski, I. E., On the length of critical orbits of stable quadratic polynomials, Proc. Amer. Math. Soc. 138 (2010), 2653–2656.CrossRefGoogle Scholar
16.Stichtenoth, H., Algebraic function fields and codes (Springer-Verlag, Berlin, 1993).Google Scholar