On stable quadratic polynomials

Omran Ahmadi*, Florian Luca, Alina Ostafe, Igor E. Shparlinski

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

10 Citations (Scopus)


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.

Original languageEnglish
Pages (from-to)359-369
Number of pages11
JournalGlasgow Mathematical Journal
Issue number2
Publication statusPublished - May 2012


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