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How many zeros of a random polynomial are real? (English) Zbl 0820.34038

The authors 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. They show that the expected number of real zeros is simply the length of the moment curve \((1,t,\dots, t^ n)\) projected onto the surface of the unit sphere, devided by \(\pi\). Also the authors consider random \(n\)th degree polynomials with independent normally distributed coefficients, each with mean zero, but with the variance of the \(i\)th coefficient equal to \(\left(\begin{smallmatrix} n\\ i\end{smallmatrix}\right)\).
It has \(E_ n= \sqrt n\) real zeros on average. Then the authors relax Kac’s assumptions by considering a variety of random sums, series and distributions.

MSC:

34F05 Ordinary differential equations and systems with randomness
30B20 Random power series in one complex variable
60D05 Geometric probability and stochastic geometry

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[1] Orlando Alvarez, Enzo Marinari, and Paul Windey , Random surfaces and quantum gravity, NATO Advanced Science Institutes Series B: Physics, vol. 262, Plenum Press, New York, 1991. · Zbl 0745.00026
[2] A. T. Bharucha-Reid and M. Sambandham, Random polynomials, Probability and Mathematical Statistics, Academic Press, Inc., Orlando, FL, 1986. · Zbl 0615.60058
[3] A. Bloch and G. Pólya, On the roots of certain algebraic equations, Proc. London Math. Soc. (3) 33 (1932), 102-114. · JFM 57.0128.03
[4] E. Bogomolny, O. Bohigas, and P. Lebœuf, Distribution of roots of random polynomials, Phys. Rev. Lett. 68 (1992), no. 18, 2726 – 2729. · Zbl 0969.81540
[5] T. Bonnesen and W. Fenchel, Theory of convex bodies, BCS Associates, Moscow, ID, 1987. Translated from the German and edited by L. Boron, C. Christenson and B. Smith. · Zbl 0628.52001
[6] Ralph Philip Boas Jr., Entire functions, Academic Press Inc., New York, 1954. · Zbl 0058.30201
[7] Paul A. Clement, A class of triple-diagonal matrices for test purposes, SIAM Rev. 1 (1959), 50 – 52. · Zbl 0117.14202
[8] Persi Diaconis, Group representations in probability and statistics, Institute of Mathematical Statistics Lecture Notes — Monograph Series, vol. 11, Institute of Mathematical Statistics, Hayward, CA, 1988. · Zbl 0695.60012
[9] Persi Diaconis, R. L. Graham, and J. A. Morrison, Asymptotic analysis of a random walk on a hypercube with many dimensions, Random Structures Algorithms 1 (1990), no. 1, 51 – 72. · Zbl 0723.60085
[10] Alan Edelman, Eigenvalues and condition numbers of random matrices, SIAM J. Matrix Anal. Appl. 9 (1988), no. 4, 543 – 560. · Zbl 0678.15019
[11] -, Eigenvalues and condition numbers of random matrices, Ph.D. thesis, Department of Mathematics, MIT, 1989.
[12] -, Math 273e: Eigenvalues of random matrices, Personal Lecture Notes, University of California, Berkeley, Spring 1993, unpublished.
[13] Alan Edelman, Eric Kostlan, and Michael Shub, How many eigenvalues of a random matrix are real?, J. Amer. Math. Soc. 7 (1994), no. 1, 247 – 267. · Zbl 0790.15017
[14] A. Edelman, Eigenvalue roulette and random test matrices, Numerical Linear Algebra, Digital Signal Processing and Parallel Algorithms , NATO Adv. Sci. Inst. Ser. F: Comput. Systems Sci., vol. 70, Springer-Verlag, New York, 1991, pp. 485-488.
[15] -, The circular law and the probability that a random matrix has k real eigenvalues, preprint.
[16] P. Erdös and P. Turán, On the distribution of roots of polynomials, Ann. of Math. (2) 51 (1950), 105-119. · Zbl 0036.01501
[17] Kambiz Farahmand, On the average number of real roots of a random algebraic equation, Ann. Probab. 14 (1986), no. 2, 702 – 709. · Zbl 0609.60074
[18] J. Faraut and A. Koranyi, Analysis on symmetric cones, draft of book. · Zbl 0718.32026
[19] V. L. Girko, Theory of random determinants, Mathematics and its Applications (Soviet Series), vol. 45, Kluwer Academic Publishers Group, Dordrecht, 1990. Translated from the Russian.
[20] I. P. Goulden and D. M. Jackson, Combinatorial constructions for integrals over normally distributed random matrices, Proc. Amer. Math. Soc. 123 (1995), no. 4, 995 – 1003. · Zbl 0836.05076
[21] I. Gradshteyn and I. Ryzhik, Table of integrals, series, and products, Academic Press, New York, 1980. · Zbl 0521.33001
[22] Phillip Griffiths and Joseph Harris, Principles of algebraic geometry, Wiley-Interscience [John Wiley & Sons], New York, 1978. Pure and Applied Mathematics. · Zbl 0408.14001
[23] J. M. Hammersley, The zeros of a random polynomial, Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1954 – 1955, vol. II, University of California Press, Berkeley and Los Angeles, 1956, pp. 89 – 111.
[24] Philip J. Hanlon, Richard P. Stanley, and John R. Stembridge, Some combinatorial aspects of the spectra of normally distributed random matrices, Hypergeometric functions on domains of positivity, Jack polynomials, and applications (Tampa, FL, 1991) Contemp. Math., vol. 138, Amer. Math. Soc., Providence, RI, 1992, pp. 151 – 174. · Zbl 0789.05092
[25] N. Higham, The test matrix toolbox for Matlab, Numerical Analysis Report No. 237, University of Manchester, England, December 1993.
[26] M. Kac, On the average number of real roots of a random algebraic equation, Bull. Amer. Math. Soc. 49 (1943), 314 – 320. · Zbl 0060.28602
[27] M. Kac, On the average number of real roots of a random algebraic equation. II, Proc. London Math. Soc. (2) 50 (1949), 390 – 408. · Zbl 0033.14702
[28] Jean-Pierre Kahane, Some random series of functions, D. C. Heath and Co. Raytheon Education Co., Lexington, Mass., 1968. · Zbl 0805.60007
[29] S. Kobayashi and K. Nomizu, Foundations of differential geometry, Volume 2, Wiley, New York, 1969. · Zbl 0175.48504
[30] E. Kostlan, Random polynomials and the statistical fundamental theorem of algebra, unpublished, 1987.
[31] E. Kostlan, On the distribution of roots of random polynomials, From Topology to Computation: Proceedings of the Smalefest (Berkeley, CA, 1990) Springer, New York, 1993, pp. 419 – 431. · Zbl 0788.60069
[32] -, On the expected number of real roots of a system of random polynomial equations, in preparation.
[33] -, On the expected volume of a real algebraic variety, in preparation.
[34] J. Littlewood and A. Offord, On the number of real roots of a random algebraic equation, J. London Math. Soc. 13 (1938), 288-295. · Zbl 0020.13604
[35] Froim Marcus, On the metrics of G. Fubini, Proceedings of the mathematical congress in celebration of the one hundredth birthday of Guido Fubini and Francesco Severi (Turin, 1979), 1981, pp. 235 – 242 (1982). · Zbl 0562.53016
[36] N. B. Maslova, The distribution of the number of real roots of random polynomials, Teor. Verojatnost. i Primenen. 19 (1974), 488 – 500 (Russian, with English summary).
[37] Madan Lal Mehta, Random matrices, 2nd ed., Academic Press, Inc., Boston, MA, 1991. · Zbl 0780.60014
[38] Robb J. Muirhead, Aspects of multivariate statistical theory, John Wiley & Sons, Inc., New York, 1982. Wiley Series in Probability and Mathematical Statistics. · Zbl 0556.62028
[39] C. M. Newman, Lyapunov exponents for some products of random matrices: exact expressions and asymptotic distributions, Random matrices and their applications (Brunswick, Maine, 1984) Contemp. Math., vol. 50, Amer. Math. Soc., Providence, RI, 1986, pp. 121 – 141.
[40] A. M. Odlyzko and B. Poonen, Zeros of polynomials with 0,1 coefficients, Enseign. Math. (2) 39 (1993), no. 3-4, 317 – 348. · Zbl 0814.30006
[41] S. O. Rice, The Distribution of the Maxima of a Random Curve, Amer. J. Math. 61 (1939), no. 2, 409 – 416. · Zbl 0020.38102
[42] S. O. Rice, Mathematical analysis of random noise, Bell System Tech. J. 24 (1945), 46 – 156. · Zbl 0063.06487
[43] Luis A. Santaló, Integral geometry and geometric probability, Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1976. With a foreword by Mark Kac; Encyclopedia of Mathematics and its Applications, Vol. 1. · Zbl 0342.53049
[44] Rolf Schneider, Convex bodies: the Brunn-Minkowski theory, Encyclopedia of Mathematics and its Applications, vol. 44, Cambridge University Press, Cambridge, 1993. · Zbl 0798.52001
[45] A. N. Shiryayev, Probability, Graduate Texts in Mathematics, vol. 95, Springer-Verlag, New York, 1984. Translated from the Russian by R. P. Boas.
[46] M. Shub and S. Smale, Complexity of Bezout’s theorem. II. Volumes and probabilities, Computational algebraic geometry (Nice, 1992) Progr. Math., vol. 109, Birkhäuser Boston, Boston, MA, 1993, pp. 267 – 285. · Zbl 0851.65031
[47] Herbert Solomon, Geometric probability, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1978. Ten lectures given at the University of Nevada, Las Vegas, Nev., June 9 – 13, 1975; Conference Board of the Mathematical Sciences — Regional Conference Series in Applied Mathematics, No. 28. · Zbl 0382.60016
[48] M. Spivak, A comprehensive introduction to differential geometry, 2nd ed., Publish or Perish, Berkeley, CA, 1979. · Zbl 0439.53001
[49] Olga Taussky and John Todd, Another look at a matrix of Mark Kac, Proceedings of the First Conference of the International Linear Algebra Society (Provo, UT, 1989), 1991, pp. 341 – 360. · Zbl 0727.15010
[50] D. V. Voiculescu, K. J. Dykema, and A. Nica, Free random variables, CRM Monograph Series, vol. 1, American Mathematical Society, Providence, RI, 1992. A noncommutative probability approach to free products with applications to random matrices, operator algebras and harmonic analysis on free groups. · Zbl 0795.46049
[51] Elias Wegert and Lloyd N. Trefethen, From the Buffon needle problem to the Kreiss matrix theorem, Amer. Math. Monthly 101 (1994), no. 2, 132 – 139. · Zbl 0799.30002
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