Abstract
The problem of finding the probability distribution of the number of zeros in some real interval of a random polynomial whose coefficients have a given continuous joint density function is considered. An algorithm which enables one to express this probability as a multiple integral is presented. Formulas for the number of zeros of random quadratic polynomials and random polynomials of higher order, some coefficients of which are non-random and equal to zero, are derived via use of the algorithm. Finally, the applicability of these formulas in numerical calculations is illustrated.
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References
A. T. Bharucha-Reid and M. Sambandham, Random Polynomials, Academic Press: Orlando/London, 1986.
A. Edelman and E. Kostlan, “How many zeros of a random polynomial are real?,” Bull. Amer. Math. Soc. (N.S.) vol. 32 pp. 1–37, 1995.
P. Erdös and A. C. Offord, “On the number of real roots of a random algebraic equation,” Proc. London Math. Soc. (3) vol. 6 pp. 139–160, 1956.
P. Erdös and P. Turán, “On the distribution of roots of polynomials,” Ann. Math. (2) vol. 51 pp. 105–119, 1950.
W. B. Fairley, Roots of Random Polynomials, Ph.D. dissertation, Harvard University: Cambridge, Mass., 1968.
J. W. Hamblen, “Distribution of roots of quadratic equations with random coefficients,” Ann. Math. Stats. vol. 27 pp. 1136–1143, 1956.
J. Hammersley, “The zeros of a random polynomial,” Proc. Third Berkeley Symposium on Math. Stats. and Probability, vol. II, University of California Press, 1956, pp. 89–111.
L. A. Shepp and R. J. Vanderbei, “The complex zeros of random polynomials,” Trans. Amer. Math. Soc. vol. 347 pp. 4365–4384, 1995.
B. Shiffman and S. Zelditch, “Equilibrium distribution of zeros of random polynomials,” International Math. Res. Notes vol. 2003 pp. 25–49, 2003.
E. Shmerling, Linear Systems with Random Coefficients, Ph.D. dissertation, Bar-Ilan University, Israel, 2000.
E. Shmerling and K. J. Hochberg, “Asymptotic behavior of roots of random polynomial equations,” Proc. Amer. Math. Soc. vol. 130 pp. 2761–2770, 2002.
M. Sodin, “Zeros of Gaussian analytic functions,” Math. Res. Lett. vol. 7 pp. 371–381, 2000.
B. L. van der Waerden, Algebra I, Springer Verlag: New York, 1971.
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Shmerling, E., Hochberg, K.J. Sturm's Method in Counting Roots of Random Polynomial Equations. Methodology and Computing in Applied Probability 6, 203–218 (2004). https://doi.org/10.1023/B:MCAP.0000017713.58934.d3
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DOI: https://doi.org/10.1023/B:MCAP.0000017713.58934.d3