Skip to main content
×
Home
    • Aa
    • Aa

ON STABLE QUADRATIC POLYNOMIALS

  • OMRAN AHMADI (a1), FLORIAN LUCA (a2), ALINA OSTAFE (a3) and IGOR E. SHPARLINSKI (a4)
Abstract
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.

  • View HTML
    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      ON STABLE QUADRATIC POLYNOMIALS
      Available formats
      ×
      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about sending content to Dropbox.

      ON STABLE QUADRATIC POLYNOMIALS
      Available formats
      ×
      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about sending content to Google Drive.

      ON STABLE QUADRATIC POLYNOMIALS
      Available formats
      ×
Copyright
Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

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.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Glasgow Mathematical Journal
  • ISSN: 0017-0895
  • EISSN: 1469-509X
  • URL: /core/journals/glasgow-mathematical-journal
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Keywords:

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 19 *
Loading metrics...

Abstract views

Total abstract views: 60 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 24th September 2017. This data will be updated every 24 hours.