The complex zeros of random polynomials

Authors:
Larry A. Shepp and Robert J. Vanderbei

Journal:
Trans. Amer. Math. Soc. **347** (1995), 4365-4384

MSC:
Primary 30C15; Secondary 60G99

DOI:
https://doi.org/10.1090/S0002-9947-1995-1308023-8

MathSciNet review:
1308023

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Mark Kac gave an explicit formula for the expectation of the number, , of zeros of a random polynomial,

*reals*. Here, are independent standard normal random variables. In fact, for each , he obtained an explicit intensity function for which

**[1]**Lars V. Ahlfors,*Complex analysis*, 3rd ed., McGraw-Hill Book Co., New York, 1978. An introduction to the theory of analytic functions of one complex variable; International Series in Pure and Applied Mathematics. MR**510197****[2]**A. T. Bharucha-Reid and M. Sambandham,*Random polynomials*, Probability and Mathematical Statistics, Academic Press, Inc., Orlando, FL, 1986. MR**856019****[3]**Alan Edelman and Eric Kostlan,*How many zeros of a random polynomial are real?*, Bull. Amer. Math. Soc. (N.S.)**32**(1995), no. 1, 1–37. MR**1290398**, https://doi.org/10.1090/S0273-0979-1995-00571-9**[4]**P. Erdös and P. Turán,*On the distribution of roots of polynomials*, Ann. of Math. (2)**51**(1950), 105–119. MR**0033372**, https://doi.org/10.2307/1969500**[5]**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. MR**0084888****[6]**I. A. Ibragimov and N. B. Maslova,*The mean number of real zeros of random polynomials. I. Coefficients with zero mean*, Teor. Verojatnost. i Primenen.**16**(1971), 229–248 (Russian, with English summary). MR**0286157****[7]**M. Kac,*On the average number of real roots of a random algebraic equation*, Bull. Amer. Math. Soc.**49**(1943), 314–320. MR**0007812**, https://doi.org/10.1090/S0002-9904-1943-07912-8**[8]**-,*Probability and related topics in physical sciences*, Interscience, London, 1959.**[9]**S. V. Konyagin,*On the minimum modulus of random trigonometric polynomials with coefficients ±1*, Mat. Zametki**56**(1994), no. 3, 80–101, 158 (Russian, with Russian summary); English transl., Math. Notes**56**(1994), no. 3-4, 931–947 (1995). MR**1309842**, https://doi.org/10.1007/BF02362411**[10]**B. F. Logan and L. A. Shepp,*Real zeros of random polynomials*, Proc. London Math. Soc. (3)**18**(1968), 29–35. MR**0234512**, https://doi.org/10.1112/plms/s3-18.1.29**[11]**B. F. Logan and L. A. Shepp,*Real zeros of random polynomials. II*, Proc. London Math. Soc. (3)**18**(1968), 308–314. MR**0234513**, https://doi.org/10.1112/plms/s3-18.2.308**[12]**S. O. Rice,*Mathematical analysis of random noise*, Bell System Tech. J.**24**(1945), 46–156. MR**0011918**, https://doi.org/10.1002/j.1538-7305.1945.tb00453.x**[13]**D. C. Stevens,*The average and variance of the number of real zeros of random functions*, Ph.D. Dissertation, New York University, New York, 1965.**[14]**D. I. Šparo and M. G. Šur,*On the distribution of roots of random polynomials*, Vestnik Moskov. Univ. Ser. I Mat. Meh.**1962**(1962), no. 3, 40–43 (Russian, with English summary). MR**0139199**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
30C15,
60G99

Retrieve articles in all journals with MSC: 30C15, 60G99

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1995-1308023-8

Article copyright:
© Copyright 1995
American Mathematical Society