Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Pairing of zeros and critical points for random polynomials

Boris Hanin

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Let $p_{N}$ be a random degree $N$ polynomial in one complex variable whose zeros are chosen independently from a fixed probability measure $\mu$ on the Riemann sphere $S^{2}$. This article proves that if we condition $p_{N}$ to have a zero at some fixed point $\xi\in S^{2}$, then, with high probability, there will be a critical point $w_{\xi}$ at a distance $N^{-1}$ away from $\xi$. This $N^{-1}$ distance is much smaller than the $N^{-1/2}$ typical spacing between nearest neighbors for $N$ i.i.d. points on $S^{2}$. Moreover, with the same high probability, the argument of $w_{\xi}$ relative to $\xi$ is a deterministic function of $\mu$ plus fluctuations on the order of $N^{-1}$.


Soit $p_{N}$ un polynôme aléatoire de degré $N$ en une variable complexe tel que ses zéros sont distribués indépendamment suivant une mesure de probabilité $\mu$ fixée et définie sur la sphère de Riemann $S^{2}$. Cet article prouve que si nous conditionnons $p_{N}$ pour avoir un zéro en un point fixé $\xi\in S^{2}$, alors, avec grande probabilité, il y aura un point critique $w_{\xi}$ à une distance $N^{-1}$ de $\xi$. Cette distance $N^{-1}$ est beaucoup plus petite que l’espacement typique entre deux points voisins pour $N$ points i.i.d. sur $S^{2}$, qui lui est d’ordre $N^{-1/2}$. De plus, avec la méme grande probabilité, l’argument de $w_{\xi}$ relativement à $\xi$ est une fonction déterministe de $\mu$, plus des fluctuations d’ordre $N^{-1}$.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 53, Number 3 (2017), 1498-1511.

Received: 3 February 2016
Revised: 12 April 2016
Accepted: 22 May 2016
First available in Project Euclid: 21 July 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 30C10: Polynomials 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} 60G60: Random fields

Zeros Critical points Random polynomials Gauss–Lucas


Hanin, Boris. Pairing of zeros and critical points for random polynomials. Ann. Inst. H. Poincaré Probab. Statist. 53 (2017), no. 3, 1498--1511. doi:10.1214/16-AIHP767.

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  • [1] P. Abbott and B. Torrence. Sendov’s conjecture: Wolfram demonstration project. Available at
  • [2] G. Anderson, A. Guionnet and O. Zeitouni. An Introduction to Random Matrices. Cambridge Studies in Advanced Mathematics 118. Cambridge University Press, Cambridge, 2010.
  • [3] T. Bloom. Random polynomials and Green functions. Int. Math. Res. Not. IMRN 28 (2005) 1689–1708.
  • [4] T. Bloom and B. Shiffman. Zeros of random polynomials on $\mathbb{C}^{m}$. Math. Res. Lett. 14 (2007) 469–479.
  • [5] M. Dennis and J. Hannay. Saddle points in the chaotic analytic function and Ginibre characteristic polynomial. J. Phys. A 36 (2003) 3379–3384. (Special issue ‘Random Matrix Theory’).
  • [6] Y. Do, O. Nguyen and V. Vu. Roots of random polynomials with arbitrary coefficients. Preprint. Available at
  • [7] O. Feldheim and A. Sen. Double roots of random polynomials with integer coefficients. Preprint. Available at
  • [8] R. Feng. Conditional expectations of random holomorphic fields on Riemann surfaces. Int. Math. Res. Not. IMRN. To appear, 2017. DOI:10.1093/imrn/rnw152.
  • [9] B. Hanin. Correlations and pairing between zeros and critical points of Gaussian random polynomials. Int. Math. Res. Not. IMRN 2 (2015) 381–421.
  • [10] B. Hanin. Pairing of zeros and critical points for random meromorphic functions on Riemann surfaces. Math. Res. Lett. 22 (1) (2015) 111–140.
  • [11] R. Hough, M. Krishnapur, Y. Peres and B. Virág. Zeros of Gaussian Analytic Functions and Determinantal Point Processes. University Lecture Series 51. American Mathematical Society, Providence, RI, 2009.
  • [12] Z. Kabluchko. Critical points of random polynomials with independent and identically distributed roots. Proc. Amer. Math. Soc. 143 (2) (2015) 695–702.
  • [13] M. Marden. The Geometry of the Zeros of a Polynomial in a Complex Variable. American Mathematical Society, New York, 1949.
  • [14] F. Nazarov, M. Sodin and A. Volberg. Transportation to random zeroes by the gradient flow. Geom. Funct. Anal. 17 (2007) 887–935.
  • [15] R. Peled, A. Sen and O. Zeitouni. Double roots of random Littlewood polynomials. Israel J. Math. 213 (2016) 55–77.
  • [16] R. Pemantle and I. Rivlin. The distribution of the zeroes of the derivative of a random polynomial. In Advances in Combinatorics 259–273. I. S. Kotsireas and E. V. Zima (Eds). Springer, Heidelberg, 2013.
  • [17] B. Shiffman and S. Zelditch. Equilibrium distribution of zeros of random polynomials. Int. Math. Res. Not. IMRN 1 (2003) 25–49.
  • [18] M. Sodin. Zeroes of Gaussian analytic functions. Math. Res. Lett. 7 (2000) 371–381.
  • [19] S. Subramanian. On the distribution of critical points of a polynomial. Electron. Commun. Probab. 17 (37) (2012) 1–9.
  • [20] E. Titchmarsh. The Theory of Functions. Oxford University Press, Oxford, 1939.