Illinois Journal of Mathematics

The expected number of complex zeros of complex random polynomials

Katrina Ferrier, Micah Jackson, Andrew Ledoan, Dhir Patel, and Huong Tran

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Abstract

By using the technique introduced in 1995 by Shepp and Vanderbei, we derive an exact formula for the expected number of complex zeros of a complex random polynomial due to Kac. The explicit evaluation of the average intensity function is obtained in closed form in the case of standard normal coefficients. In addition, we provide the limiting expressions for the intensity function and the expected number of zeros in open circular disks in the complex plane.

Article information

Source
Illinois J. Math., Volume 61, Number 1-2 (2017), 211-224.

Dates
Received: 17 April 2017
Revised: 21 August 2017
First available in Project Euclid: 3 March 2018

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1520046216

Digital Object Identifier
doi:10.1215/ijm/1520046216

Mathematical Reviews number (MathSciNet)
MR3770843

Zentralblatt MATH identifier
06864466

Subjects
Primary: 30C15: Zeros of polynomials, rational functions, and other analytic functions (e.g. zeros of functions with bounded Dirichlet integral) {For algebraic theory, see 12D10; for real methods, see 26C10} 30B20: Random power series 26C10: Polynomials: location of zeros [See also 12D10, 30C15, 65H05] 60B99: None of the above, but in this section

Citation

Ferrier, Katrina; Jackson, Micah; Ledoan, Andrew; Patel, Dhir; Tran, Huong. The expected number of complex zeros of complex random polynomials. Illinois J. Math. 61 (2017), no. 1-2, 211--224. doi:10.1215/ijm/1520046216. https://projecteuclid.org/euclid.ijm/1520046216


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References

  • R. J. Adler, The geometry of random fields, Wiley Ser. Probab. Math. Statist., John Wiley & Sons, Ltd., Chichester, 1981.
  • A. T. Bharucha-Reid and M. Sambandham, Random polynomials, Probab. Math. Statist., Academic Press, Inc., Orlando, Florida, 1986.
  • A. Edelman and E. Kostlan, How many zeros of a random polynomial are equal?, Bull. Amer. Math. Soc. (N.S.) 32 (1995), no. 1, 1–37.
  • A. Edelman and E. Kostlan, Erratum: “How many zeros of a random polynomial are real?” (Bull. Amer. Math. Soc. (N.S.) 32 (1995), no. 1, 1–37), Bull. Amer. Math. Soc. (N.S.) 33 (1996), no. 3, 325.
  • K. Farahmand, Complex roots of a random algebraic polynomial, J. Math. Anal. Appl. 210 (1997), no. 2, 724–730.
  • K. Farahmand, Topics in random polynomials, Pitman Res. Notes Math. Ser., vol. 393, Addison-Wesley, Longman, Edinburgh Gate, Harlow, 1998.
  • J. M. Hammersley, The zeros of a random polynomial, Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1954–1955, vol. II, University of California Press, Berkeley and Los Angeles, 1956, pp. 89–111.
  • J. B. Hough, M. Krishnapur, Y. Peres and B. Virág, Zeros of Gaussian analytic functions and determinantal point processes, Univ. Lect. Ser., vol. 51, Amer. Math. Soc., Providence, Rhode Island, 2009.
  • I. Ibragimov and O. Zeitouni, On roots of random polynomials, Trans. Amer. Math. Soc. 349 (1997), no. 6, 2427–2441.
  • M. Kac, On the average number of real roots of a random algebraic equation, Bull. Amer. Math. Soc. 49 (1943), 314–320.
  • M. Kac, A correction to “On the average number of real roots of a random algebraic equation” (Bull. Amer. Math. Soc. 49 (1943), 314–320), Bull. Amer. Math. Soc. 49 (1943), 938.
  • A. Ledoan, Explicit formulas for the distribution of complex zeros of a family of random sums, J. Math. Anal. Appl. 444 (2016), no. 2, 1304–1320.
  • J. E. Littlewood and A. C. Offord, On the number of real roots of a random algebraic equation, J. London Math. Soc. 13 (1938), no. 4, 288–295.
  • L. A. Shepp and R. J. Vanderbei, The complex zeros of random polynomials, Trans. Amer. Math. Soc. 347 (1995), no. 11, 4365–4384.
  • R. J. Vanderbei, The complex zeros of random sums, preprint, 2015; available at https://arxiv.org/pdf/1508.05162.pdf.
  • A. M. Yeager, Zeros of random linear combinations of entire functions with complex Gaussian coefficients, preprint, 2016; available at https://arxiv.org/pdf/1605.06836.pdf.