How many zeros of a random polynomial are real

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, . . . , tn) projected onto the surface of the unit sphere, divided by π. 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.

[1]  M. Kac.  A correction to “On the average number of real roots of a random algebraic equation” , 1943 .

[2]  Paul Erdös,et al.  On the Distribution of Roots of Polynomials , 1950 .

[3]  A. Edelman.  Eigenvalue Roulette and Random Test Matrices , 1993 .

[4]  George Polya,et al.  On the Roots of Certain Algebraic Equations , 1932 .

[5]  M. Kac,et al.  On the Average Number of Real Roots of a Random Algebraic Equation (II) , 1948 .

[6]  P. Diaconis.  Group representations in probability and statistics , 1988 .

[7]  J. Faraut,et al.  Analysis on Symmetric Cones , 1995 .

[8]  K. Farahmand.  On the Average Number of Real Roots of a Random Algebraic Equation , 1986 .

[9]  Ian P. Goulden,et al.  Combinatorial constructions for integrals over normally distributed random matrices , 1995 .

[10]  Ronald L. Graham,et al.  Asymptotic Analysis of a Random Walk on a Hypercube with Many Dimensions , 1990, Random Struct. Algorithms.

[11]  J. Kahane.  Some Random Series of Functions , 1985 .

[12]  A. Edelman,et al.  How many eigenvalues of a random matrix are real , 1994 .

[13]  E. Kostlan.  On the Distribution of Roots of Random Polynomials , 1993 .

[14]  S. Rice.  The Distribution of the Maxima of a Random Curve , 1939 .

[15]  Vi︠a︡cheslav Leonidovich Girko,et al.  Theory of random determinants , 1990 .

[16]  K. Nomizu,et al.  Foundations of Differential Geometry , 1963 .

[17]  M. Spivak.  A comprehensive introduction to differential geometry , 1979 .

[18]  N. Higham.  The Test Matrix Toolbox for MATLAB , 1993 .

[19]  Stanislaw J. Szarek,et al.  Condition numbers of random matrices , 1991, J. Complex..

[20]  E. Marinari,et al.  Random surfaces and quantum gravity , 1991 .

[21]  A. Edelman.  Eigenvalues and condition numbers of random matrices , 1988 .

[22]  R. Schneider.  Convex Bodies: The Brunn–Minkowski Theory: Minkowski addition , 1993 .

[23]  Alexandru Nica,et al.  Free random variables : a noncommutative probability approach to free products with applications to random matrices, operator algebras, and harmonic analysis on free groups , 1992 .

[24]  I. S. Gradshteyn,et al.  Table of Integrals, Series, and Products , 1976 .

[25]  R. Muirhead.  Aspects of Multivariate Statistical Theory , 1982, Wiley Series in Probability and Statistics.

[26]  S. Smale,et al.  Complexity of Bezout’s Theorem II Volumes and Probabilities , 1993 .

[27]  N. Maslova.  On the Distribution of the Number of Real Roots of Random Polynomials , 1975 .

[28]  Joe W. Harris,et al.  Principles of Algebraic Geometry , 1978 .

[29]  Charles M. Newman,et al.  Lyapunov exponents for some products of random matrices: Exact expressions and asymptotic distributions , 1986 .

[30]  Leboeuf,et al.  Distribution of roots of random polynomials. , 1992, Physical review letters.

[31]  L. Trefethen,et al.  From the Buffon Needle Problem to the Kreiss Matrix Theorem , 1994 .

[32]  J. M. Hammersley,et al.  The Zeros of a Random Polynomial , 1956 .

[33]  Alexandru Nica,et al.  Free random variables , 1992 .

[34]  J. Littlewood,et al.  On the Number of Real Roots of a Random Algebraic Equation , 1938 .

[35]  P. Clement,et al.  A Class of Triple-Diagonal Matrices for Test Purposes , 1959 .

[36]  Leo F. Boron,et al.  Theory of Convex Bodies , 1988 .

[37]  John Todd,et al.  Another look at a matrix of Mark Kac , 1991 .

[38]  S. Rice.  Mathematical analysis of random noise , 1944 .