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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.

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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.

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

Snapshots

Details

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

References

[1] G. Peyré. "Oldies but Goldies: J. Hammersley, The Zeros of a Random Polynomial, 1956." (Aug 15, 2019) twitter.com/gabrielpeyre/status/1158241298303901696.

[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. projecteuclid.org/euclid.bsmsp/1200502008.

[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. math.iisc.ernet.in/~manju/GAF_book.pdf.

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