Skip to main content
SpringerLink
Log in

Finite polynomial orbits in quadratic rings

  • Published:
The Ramanujan Journal Aims and scope Submit manuscript

Abstract

All finite orbits of polynomial mappings in one variable in rings of integers of quadratic number fields are determined. The largest such orbit has seven elements and lies in the third cyclotomic field.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Baron, G.: Polynomiteration in algebraischen Zahlkörpern, preprint (1990)

  2. Batut, C., Belabas, K., Bernardi, D., Cohen, H., Olivier, M.: User’s Guide to PARI/GP version 2.1.0, Talence 2000 (The program PARI is available at the web site www.parigp-home.de)

  3. Boduch, J.: Polynomial cycles in rings of algebraic integers M.A. thesis, Wrocł aw University (1990) (Polish)

  4. Daberkow, M., Fieker, C., Klüners, J., Pohst, M., Roegner, K., Wildanger, K.: Kant V4, Journal of Symb Comp. 24, 267–283 (1997)

    Google Scholar 

  5. Ennola, V.: Cubic fields with exceptional units. Computational Number Theory de Gruyter, pp. 104–128 (1991)

  6. Flynn, E.V., Poonen, B., Schaefer, E.F.: Cycles of quadratic polynomials and rational points on a genus 2-curve. Duke Math. J. 90, 435–463 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  7. Halter-Koch, F, Narkiewicz, W.: Polynomial cycles and dynamical units. Proc. Conf. Analytic and Elementary Number Theory Wien, pp. 70–80 (1997)

  8. Halter-Koch, F, Narkiewicz, W.: Scarcity of finite polynomial orbits. Publicat. Math. (Debrecen) 56, 405–414 (2000)

    MATH  MathSciNet  Google Scholar 

  9. Morton, P.: Arithmetic properties of periodic points of quadratic maps, II. Acta Arith. 82, 89–102 (1998)

    MathSciNet  Google Scholar 

  10. Morton, P., Silverman, J.: Rational periodic points of rational functions Internat. Math. Res. Notices 97–110 (1994)

  11. Nagell, T.: Les points exceptionnels rationnels sur certaines cubiques du premier genre. Acta Arith. 5, 333–357 (1954)

    MathSciNet  Google Scholar 

  12. Narkiewicz, W.: Polynomial cycles in algebraic number fields. Colloq. Math 58, 149–153 (1989)

    MathSciNet  Google Scholar 

  13. Narkiewicz, W.: Polynomial Mappings Lecture Notes in Math. 1600, Sprin-ger (1995)

  14. Narkiewicz, W., Pezda, T.: Finite polynomial orbits in finitely generated domains. Monatsh. Math. 124, 309–316 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  15. Northcott, D.G.: Periodic points on an algebraic variety. Ann. Math. 51, 167–177 (1950)

    Article  MATH  MathSciNet  Google Scholar 

  16. Pezda, T.: Polynomial cycles in certain local domains. Acta Arith. 66, 11–22 (1994)

    MATH  MathSciNet  Google Scholar 

  17. Pezda, T.: Cycles of polynomial mappings in several variables. Manuscr. Math. 83, 279–289 (1994)

    MATH  MathSciNet  Google Scholar 

  18. Pezda, T.: Cycles of polynomial mappings in several variables over rings of integers in finite extensions of the rationals. Acta Arith. 108, 127–146 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  19. Poonen, B.: The classification of rational preperiodic points of quadratic polynomials over Q: a refined conjecture. Math. Zeitschr. 228, 11–29 (1998)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics, Wrocław University, Plac Grunwaldzki 2–4, PL-50-384, Wrocław, Poland

    Roman Marszałek & Władyslaw Narkiewicz

Authors
  1. Roman Marszałek
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Władyslaw Narkiewicz
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Roman Marszałek.

Additional information

2000 Mathematics Subject Classification Primary—11R09, 11R11, 37F10

Rights and permissions

Reprints and permissions

About this article

Cite this article

Marszałek, R., Narkiewicz, W. Finite polynomial orbits in quadratic rings. Ramanujan J 12, 91–130 (2006). https://doi.org/10.1007/s11139-006-9578-z

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11139-006-9578-z

Keywords

Search

Navigation