Glasgow Mathematical Journal

Research Article

ON STABLE QUADRATIC POLYNOMIALS

OMRAN AHMADIa1, FLORIAN LUCAa2, ALINA OSTAFEa3 and IGOR E. SHPARLINSKIa4

a1 Claude Shannon Institute, University College Dublin, Dublin 4, Ireland e-mail: omran.ahmadi@ucd.ie

a2 Instituto de Matemáticas, Universidad Nacional Autonoma de México, C.P. 58089, Morelia, Michoacán, Mexico e-mail: fluca@matmor.unam.mx

a3 Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190 CH-8057, Zürich, Switzerland e-mail: alina.ostafe@math.uzh.ch

a4 Department of Computing, Macquarie University, Sydney, NSW 2109, Australia e-mail: igor.shparlinski@mq.edu.au

Abstract

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.

(Received June 27 2010)

(Revised April 15 2011)

(Accepted October 10 2011)

2010 Mathematics Subject Classification

  • 11C08;
  • 11T06;
  • 37P05