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The Ramanujan Journal

, Volume 42, Issue 2, pp 363–369 | Cite as

On the number of roots of self-inversive polynomials on the complex unit circle

  • R. S. Vieira
Article

Abstract

We present a sufficient condition for a self-inversive polynomial to have a fixed number of roots on the complex unit circle. We also prove that these roots are simple when that condition is satisfied. This generalizes the condition found by Lakatos and Losonczi for all the roots of a self-inversive polynomial to lie on the complex unit circle.

Keywords

Self-inversive polynomials Self-reciprocal polynomials Salem polynomials Bethe Ansatz equations 

Mathematics Subject Classification

11K16 12D10 12E10 97I80 

Notes

Acknowledgments

The author thanks A. Lima-Santos for the motivation, comments, and discussions and also the anonymous referee of this paper for his valuable suggestions.

References

  1. 1.
    Lakatos, P., Losonczi, L.: Self-inversive polynomials whose zeros are on the unit circle. Publ. Math. Debr. 65, 409–420 (2004)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Cohn, A.: über die Anzahl der Wurzeln einer algebraischen Gleichung in einem Kreise. Math. Z. 14, 110–148 (1922)CrossRefzbMATHGoogle Scholar
  3. 3.
    Rouché, E.: Mémoire sur la série de Lagrange. J. École Polytech. 22, 217–218 (1862)Google Scholar
  4. 4.
    Bonsall, F.F., Marden, M.: Zeros of self-inversive polynomials. Proc. Am. Math. Soc. 3, 471–475 (1952)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Ancochea, G.: Zeros of self-inversive polynomials. Proc. Am. Math. Soc. 4, 900–902 (1953)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Schinzel, A.: Self-inversive polynomials with all zeros on the unit circle. Ramanujan J. 9(1–2), 19–23 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Marden, M.: Geometry of Polynomials. Mathematical Surveys. American Mathematical Society, Providence (1966)zbMATHGoogle Scholar
  8. 8.
    Bertin, M.J., Decomps-Guilloux, A., Grandet-Hugot, M., Pathiaux-Delefosse, M., Schreiber, J.P.: Pisot and Salem numbers. Birkhäuser Verlag, Basel (1992)CrossRefzbMATHGoogle Scholar
  9. 9.
    Smyth, C.: Seventy years of Salem numbers: a survey. Bull. Lond. Math. Soc. 47(3), 379–395 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Boyd, D.W.: Pisot and Salem numbers in intervals of the real line. Math. Comput. 32, 1244–1260 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Boyd, D.W.: Small Salem numbers. Duke Math. J. 44, 315–328 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Hironaka, E.: What is..Lehmer’s number? Not. Am. Math. Soc. 56(3), 374–375 (2009)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Vieira, R.S., Lima-Santos, A.: Where are the roots of the Bethe Ansatz equations? Phys. Lett. A 379(37), 2150–2153 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Bethe, H.: Zur Theorie der Metalle I. Z. Phys. 71, 205–226 (1931)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Departamento de FísicaUniversidade Federal de São CarlosSão CarlosBrazil

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