Electronic Journal of Probability

Limiting empirical distribution of zeros and critical points of random polynomials agree in general

Tulasi Ram Reddy

Full-text: Open access


In this article, we study critical points (zeros of derivative) of random polynomials. Take two deterministic sequences $\{a_n\}_{n\geq 1}$ and $\{b_n\}_{n\geq 1}$ of complex numbers whose limiting empirical measures are the same. By choosing $\xi _n = a_n$ or $b_n$ with equal probability, define the sequence of polynomials by $P_n(z)=(z-\xi _1)\dots (z-\xi _n)$. We show that the limiting measure of zeros and critical points agree for this sequence of random polynomials under some assumption. We also prove a similar result for triangular array of numbers. A similar result for zeros of generalized derivative (can be thought as random rational function) is also proved. Pemantle and Rivin initiated the study of critical points of random polynomials. Kabluchko proved the result considering the zeros to be i.i.d. random variables.

Article information

Electron. J. Probab., Volume 22 (2017), paper no. 74, 18 pp.

Received: 2 September 2016
Accepted: 24 July 2017
First available in Project Euclid: 13 September 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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}
Secondary: 60G57: Random measures 60B10: Convergence of probability measures

random polynomials random rational functions zeros critical points Gauss-Lucas theorem potential theory

Creative Commons Attribution 4.0 International License.


Reddy, Tulasi Ram. Limiting empirical distribution of zeros and critical points of random polynomials agree in general. Electron. J. Probab. 22 (2017), paper no. 74, 18 pp. doi:10.1214/17-EJP85. https://projecteuclid.org/euclid.ejp/1505268105

Export citation


  • [1] Cheung, P.-L., Ng, T. W., Tsai, J., and Yam, S. C. P. (2015). Higher-order, polar and Sz.-Nagy’s generalized derivatives of random polynomials with independent and identically distributed zeros on the unit circle. Comput. Methods Funct. Theory 15, 1, 159–186.
  • [2] Erdös and Turán, On a problem in the theory of uniform distribution I, I. Nederl. Akad. Wetensch. 51 (1948).
  • [3] Esseen, C. G. (1968). On the concentration function of a sum of independent random variables. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 9, 290–308.
  • [4] Forrester, P. J. (2010). Log-gases and random matrices. London Mathematical Society Monographs Series, Vol. 34. Princeton University Press, Princeton, NJ.
  • [5] Hanin, B. Pairing of zeros and critical points for random polynomials, arXiv preprint, arXiv:1601.06417.
  • [6] Hanin, B. (2015). Correlations and pairing between zeros and critical points of Gaussian random polynomials. Int. Math. Res. Not. IMRN 2, 381–421.
  • [7] Kabluchko, Z. (2015). Critical points of random polynomials with independent identically distributed roots. Proc. Amer. Math. Soc. 143, 2, 695–702.
  • [8] Sean O’Rourke, Critical points of random polynomials and characteristic polynomials of random matrices, International Mathematics Research Notices (2015), rnv331.
  • [9] Pemantle, R. and Rivin, I. (2013). The distribution of zeros of the derivative of a random polynomial. In Advances in combinatorics. Springer, Heidelberg, 259–273.
  • [10] Rahman, Q. I. and Schmeisser, G. (2002). Analytic theory of polynomials. London Mathematical Society Monographs. New Series, Vol. 26. The Clarendon Press, Oxford University Press, Oxford.
  • [11] Ransford, T. (1995). Potential theory in the complex plane. London Mathematical Society Student Texts, Vol. 28. Cambridge University Press, Cambridge.
  • [12] Reddy, T. R. On critical points of random polynomials and spectrum of certain products of random matrices, Ph.D. thesis, Indian Institute of Science, Bangalore, 2016, arXiv:1602.05298 [math.PR].
  • [13] Reeds, J. A. (1985). Asymptotic number of roots of Cauchy location likelihood equations. Ann. Statist. 13, 2, 775–784.
  • [14] Subramanian, S. On the distribution of critical points of a polynomial. Electron. Commun. Probab. 17 (2012), paper no. 37, 9 pp.
  • [15] Tao, T. and Vu, V. (2010). Random matrices: universality of ESDs and the circular law. Ann. Probab. 38, 5, 2023–2065. With an appendix by Manjunath Krishnapur.