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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
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Abstract

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

References

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