HOW MANY ZEROS OF A RANDOM POLYNOMIAL ARE REAL

被引：251

作者：

EDELMAN, A ^{[1
]}

KOSTLAN, E ^{[1
]}

机构：

[1] KAPIOLANI COMMUNITY COLL,HONOLULU,HI 96816

来源：

BULLETIN OF THE AMERICAN MATHEMATICAL SOCIETY | 1995年 / 32卷 / 01期

关键词：

RANDOM POLYNOMIALS; BUFFON NEEDLE PROBLEM; INTEGRAL GEOMETRY; RANDOM POWER SERIES; RANDOM MATRICES;

D O I：

10.1090/S0273-0979-1995-00571-9

中图分类号：

O1 [数学];

学科分类号：

0701 ; 070101 ;

摘要：

We provide an elementary geometric derivation of the Kac integral formula for the expected number of real zeros of a random polynomial with independent standard normally distributed coefficients. We show that the expected number of real zeros is simply the length of the moment curve (1,t,...,t(n)) projected onto the surface of the unit sphere, divided by pi. The probability density of the real zeros is proportional. to how fast this curve is traced out. We then relax Kac's assumptions by considering a variety of random sums, series, and distributions, and we also illustrate such ideas as integral geometry and the Fubini-Study metric.

引用

页码：1 / 37

页数：37

共 51 条

[1]

Alvarez O., Marinari E., Windey P., Random surfaces and quantum gravity, NATO Adv. Sci. Inst. Ser. B: Phys, 262, (1991)

[2]

Bharucha-Reid A.T., Sambandham M., Random Polynomials, (1986)

[3]

Bloch A., Polya G., On the roots of certain algebraic equations, Proc. London Math. Soc, 33, 3, pp. 102-114, (1932)

[4]

Bogomolny E., Bohias O., Leboeuf P., Distribution of roots of random polynomials, Phys. Rev. Lett, 68, pp. 2726-2729, (1992)

[5]

Bonnesen T., Fenchel W., Theory of Convex Bodies, (1987)

[6]

Boas R.P., Entire Functions, (1954)

[7]

Clement P.A., A class of triple-diagonal matrices for test purposes, SIAM Rev, 1, pp. 50-52, (1959)

[8]

Diaconis P., Group Representations in Probability and Statistics, (1988)

[9]

Diaconis P., Graham R.L., Morrison J.A., Asymptotic analysis of a random walk on a hypercube with many dimensions, Random Structures Algorithms, 1, pp. 51-72, (1990)

[10]

Edelman A., Eigenvalues and condition numbers of random matrices, SIAM J. Matrix Anal Appl, 9, pp. 543-560, (1988)

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