Skip to main content
SpringerLink
Log in
Book cover

Prokhorov and Contemporary Probability Theory pp 303–323Cite as

On Distribution of Zeros of Random Polynomials in Complex Plane

  • Conference paper
  • First Online:

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 33))

Abstract

Let \(G_{n}(z) = \xi _{0} + \xi _{1}z + \cdots + \xi _{n}{z}^{n}\) be a random polynomial with i.i.d. coefficients (real or complex). We show that the arguments of the roots of G n (z) are uniformly distributed in [0, 2π] asymptotically as \(n\,\rightarrow \,\infty \). We also prove that the condition \(\mathbf{E}\,\ln (1 + \vert \xi _{0}\vert )\,<\,\infty \) is necessary and sufficient for the roots to asymptotically concentrate near the unit circumference.

Mathematics Subject Classification (2010): 60-XX, 30C15

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

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  1. Hammersley, J.M.: The zeros of a random polynomial. In: Neyman, J. (ed.) Proceedings of the Third Berkeley Symposium on Mathematics, Statistics and Probability, pp. 89–111. University of California Press, Berkley (1956)

    Google Scholar 

  2. Ibragimov, I.A., Maslova, N.B.: The mean number of real zeros of random polynomials I. Coefficients with a nonzero mean. Teor. Veroyatnost. i Primenen. 16, 495–503 (1971)

    MathSciNet  Google Scholar 

  3. Ibragimov, I.A.; Maslova, N.B.: The mean number of real zeros of random polynomials II. Coefficients with a nonzero mean. Teor. Veroyatnost. i Primenen. 16, 495–503 (1971)

    MathSciNet  Google Scholar 

  4. Ibragimov, I.A., Maslova, N.B.: On the average number of real roots of random polynomials. Dokl. Akad. Nauk SSSR 199, 1004–1008 (1971)

    MathSciNet  Google Scholar 

  5. Ibragimov, I., Zeitouni, O.: On roots of random polynomials. Trans. Am. Math. Soc. 349, 2427–2441 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  6. Kac, M.: On the average number of real roots of a random algebraic equation. Bull. Am. Math. Soc. 49, 314–320 (1943)

    Article  MATH  Google Scholar 

  7. Kuipers, L., Niederreiter, H.: Uniform Distribution of Sequences. Wiley, New York (1974)

    MATH  Google Scholar 

  8. Kurosh, A.: Higher Algebra. Mir, Moscow (1988)

    Google Scholar 

  9. Logan, B.F., Shepp, L.A.: Real zeros of random polynomials. Proc. Lond. Math. Soc. 18, 29–35 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  10. Logan, B.F., Shepp, L.A.: Real zeros of random polynomials II. Proc. Lond. Math. Soc. 18, 308–314 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  11. Markushevich, A.I.: The Theory of Analytic Functions: A Brief Course. Mir, Moscow (1983)

    Google Scholar 

  12. Petrov, V.V.: Limit Theorems of Probability Theory: Sequences of Independent Random Variables. Clarendon, Oxford (1995)

    MATH  Google Scholar 

  13. Rogozin, B.A.: On the increase of dispersion of sums of independent random variables. Teor. Veroyatnost. i Primenen. 6, 106–108 (1961)

    MathSciNet  Google Scholar 

  14. Shepp, L., Farahmand, K.: Expected number of real zeros of a random polynomial with independent identically distributed symmetric long-tailed coefficients. Teor. Veroyatnost. i Primenen. 55, 196–204 (2010)

    Article  MathSciNet  Google Scholar 

  15. Shepp, L., Vanderbei, R.J.: The complex zeros of random polynomials. Trans. Am. Math. Soc. 347, 4365–4383 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  16. Shparo, D. I., Shur, M. G.: On distribution of zeros of random polynomials. Vestnik Moskov. Univ. Ser. I Mat. Mekh. 3, 40–43 (1962)

    Google Scholar 

  17. Zaporozhets, D.N.: An example of a random polynomial with unusual behavior of roots. Teor. Veroyatnost. i Primenen. 50, 549–555 (2005)

    Article  MathSciNet  Google Scholar 

  18. Zaporozhets, D.N., Nazarov, A.I.: What is the least expected number of real roots of a random polynomial? Teor. Veroyatnost. i Primenen. 53, 40–58 (2008)

    Article  Google Scholar 

Download references

Acknowledgements

A part of the work was done in the University of Bielefeld. The authors thank F. Götze for the possibility to participate at the work of CRC 701 “Spectral Structures and Topological Methods in Mathematics”. They are also grateful to A. Cole for her hospitality.

This work was partially supported by RFBR (08-01-00692, 10-01-00242), RFBR-DFG (09-0191331), NSh-4472.2010.1, and CRC 701 “Spectral Structures and Topological Methods in Mathematics”

Author information

Authors and Affiliations

  1. St. Petersburg Department of Steklov Mathematical Institute RAS, 27 Fontanka, St. Petersburg, 191023, Russia

    Ildar Ibragimov & Dmitry Zaporozhets

Authors
  1. Ildar Ibragimov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dmitry Zaporozhets
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ildar Ibragimov .

Editor information

Editors and Affiliations

  1. Steklov Mathematical Institute, Russian Academy of Sciences, Gubkina Str. 8, Moscow, 119991, Russia

    Albert N. Shiryaev

  2. Courant Institute, New York University, Mercer Street 252, New York, 10012, New York, USA

    S. R. S. Varadhan

  3. Central Economics & Math. Institute, Russian Academy of Sciences, Nakhimovsky Pr. 47, Moscow, 117418, Russia

    Ernst L. Presman

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Ibragimov, I., Zaporozhets, D. (2013). On Distribution of Zeros of Random Polynomials in Complex Plane. In: Shiryaev, A., Varadhan, S., Presman, E. (eds) Prokhorov and Contemporary Probability Theory. Springer Proceedings in Mathematics & Statistics, vol 33. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33549-5_18

Download citation

Publish with us

Policies and ethics

Search

Navigation