Probabilistic zero bounds of certain random polynomials

Sheikh, S. A.; Mir, M. I.

doi:10.2478/jamsi-2024-0004

Abstract

This paper introduces the notion of probabilistic zero bounds for random polynomials. It presents new results regarding the probabilistic bounds of random polynomials whose coefficients are independently and identically distributed as standard normal variates. Additionally, the paper provides a clear exposition of the developed methodology. To establish our results, we develop a novel approach utilizing the classical Cauchy’s bounds for the zeros of a deterministic polynomial with complex coefficients. We also corroborate our analytical results with extensive simulations. The methodology developed in the paper can potentially be applied to a broad class of problems regarding bounds and the distribution of zeros in the theory of random polynomials.

References

A. Bloch and G. Pólya, “On the roots of certain algebraic equations,” Proc. London Math. Soc., vol. 33, pp. 102–114, 1932.
Search in Google Scholar Back to article
A. T. Bharucha-Reid and M. Sambandham, Random Polynomials. Orlando, FL: Academic Press, 1986.
Search in Google Scholar Back to article
J. E. Littlewood and A. C. Offord, “On the number of real roots of a random algebraic equation,” J. Lond. Math. Soc., vol. 13, pp. 288–295, 1938.
Search in Google Scholar Back to article
J. M. Hammersley, “The zeroes of a random polynomial,” in Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, Vol. II, Berkeley: University of California Press, 1956, pp. 89–111.
Search in Google Scholar Back to article
M. Kac, “On the average number of real roots of a random algebraic equation,” Bull. Amer. Math. Soc., vol. 49, pp. 314–320, 1943.
Search in Google Scholar Back to article
A. Bloch and G. Pólya, “On the roots of certain algebraic equations,” Proc. London Math. Soc., vol. 33, pp. 102–114, 1932.
Search in Google Scholar Back to article
P. Erdős and P. Turán, “On the distribution of roots of polynomials,” Ann. of Math., vol. 51, no. 2, pp. 105–119, 1950.
Search in Google Scholar Back to article
S. O. Rice, “The distribution of the maxima of a random curve,” Amer. J. Math., vol. 61, 1939, pp. 409–416.
Search in Google Scholar Back to article
A. Edelman and E. Kostlan, “How many zeros of a random polynomial are real?,” Bull. Amer. Math. Soc., vol. 32, 1995, pp. 1–37.
Search in Google Scholar Back to article
E. Kostlan, “On the distribution of roots of random polynomials,” in From Topology to Computation: Proceedings of the Smalefest, M. W. Hirsch, J. E. Marsden, and M. Shub, Eds. New York, NY: Springer-Verlag, 1993, pp. 419–431.
Search in Google Scholar Back to article
L. A. Shepp and R. J. Vanderbei, “The complex zeros of random polynomials,” Trans. Amer. Math. Soc., 1995, pp. 4365–4384.
Search in Google Scholar Back to article
V. Thangraj and M. Sambandham, “On Random Polynomials-II: A Survey,” Neural, Parallel, and Scientific Computations, vol. 29, no. 4, 2021, pp. 230–251.
Search in Google Scholar Back to article
E. Dobriban and Z. Kabluchko, “On the distribution of complex roots of random polynomials with heavy-tailed coefficients,” SIAM Theory of Probability and its Applications, vol. 56, no. 4, 2011, pp. 537–571.
Search in Google Scholar Back to article
Q. I. Rahman and G. Schmeisser, Analytic Theory of Polynomials: Critical Points, Zeros and Extremal Properties. London Mathematical Society Monographs, 2002.
Search in Google Scholar Back to article
S. M. Ross, Introduction to Probability Models. Academic Press, 2014.
Search in Google Scholar Back to article
W. Feller, An Introduction to Probability Theory and Its Applications. John Wiley & Sons, 2008.
Search in Google Scholar Back to article
B. Datt and N. K. Govil, “On the location of the zeros of a polynomial,” J. Approx. Theory, vol. 24, 1978, pp. 78–82.
Search in Google Scholar Back to article
B. Kalantari, “An infinite family of bounds on zeros of analytic functions and relationship to Smale’s bound,” Mathematics of Computation, vol. 74, 2005, pp. 841–852.
Search in Google Scholar Back to article
M. Dehmer and A. Mowshowitz, “Bounds on the moduli of polynomial zeros,” Applied Mathematics and Computation, vol. 218, 2011, pp. 4128–4137.
Search in Google Scholar Back to article

Probabilistic zero bounds of certain random polynomials

Abstract

Paradigm

My account