Zeros of Random Kac Polynomials

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This Demonstration shows that the zeros of random Kac polynomials with independent and identically distributed (i.i.d.) coefficients cluster along the complex unit circle as the polynomial degree increases.


Setting the control "set distribution" to "normal ", the polynomial coefficients are distributed according to the standard normal distribution with zero mean and unit standard deviation [1].

Setting the control "set distribution" to "cosine of uniform ", the coefficients are the cosines of the values sampled from a uniform distribution .

You can increase the number of coefficients computed from the selected probability distribution to get a larger set of complex roots. For more than 100 or so coefficients, you can see that the zeros cluster around the unit circle in the complex plane.


Contributed by: Jessica Alfonsi (June 2020)
Open content licensed under CC BY-NC-SA



Snapshot 1: for 10 points sampled from standard normal distribution, the random polynomial zeros look scattered in the complex plane

Snapshot 2: for 50 points from uniform distribution, the random polynomial zeros begin clustering symmetrically on the complex unit circle

Snapshot 3: for 100 sampled points, the zeros appear to cluster around the complex unit circle


[1] G. Peyré. "Oldies but Goldies: J. Hammersley, The Zeros of a Random Polynomial, 1956." (Aug 15, 2019)

[2] J. M. Hammersley, "The Zeros of a Random Polynomial," in Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, Volume 2: Contributions to Probability Theory, Berkeley, CA (J. Neyman, ed.), Berkeley, CA: University of California Press, 1956 pp. 89–111.

[3] J. B. Hough, M. Krishnapur, Y. Peres and B. Virág, Zeros of Gaussian Analytic Functions and Determinantal Point Processes, Providence, RI: American Mathematical Society, 2009.

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