On roots of random polynomials

Authors:
Ildar Ibragimov and Ofer Zeitouni

Journal:
Trans. Amer. Math. Soc. **349** (1997), 2427-2441

MSC (1991):
Primary 34F05.; Secondary 26C10, 30B20

DOI:
https://doi.org/10.1090/S0002-9947-97-01766-2

MathSciNet review:
1390040

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We study the distribution of the complex roots of random polynomials of degree with i.i.d. coefficients. Using techniques related to Rice's treatment of the real roots question, we derive, under appropriate moment and regularity conditions, an exact formula for the average density of this distribution, which yields appropriate limit average densities. Further, using a different technique, we prove limit distribution results for coefficients in the domain of attraction of the stable law.

**1.**Robert J. Adler,*The geometry of random fields*, John Wiley & Sons, Ltd., Chichester, 1981. Wiley Series in Probability and Mathematical Statistics. MR**611857****2.**Ludwig Arnold,*Über die Nullstellenverteilung zufälliger Polynome*, Math. Z.**92**(1966), 12–18 (German). MR**200966**, https://doi.org/10.1007/BF01140538**3.**A. T. Bharucha-Reid and M. Sambandham,*Random polynomials*, Academic Press, New York, 1986. MR**87:60118****4.**R. N. Bhattacharya and R. Ranga Rao,*Normal approximation and asymptotic expansions*, John Wiley & Sons, New York-London-Sydney, 1976. Wiley Series in Probability and Mathematical Statistics. MR**0436272****5.**Richard Durrett,*Probability*, The Wadsworth & Brooks/Cole Statistics/Probability Series, Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA, 1991. Theory and examples. MR**1068527****6.**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**7.**P. Erdös and A. C. Offord, On the number of real roots of a random algebraic equation,*Proc. London Math. Soc.*3:139-160, 1956. MR**17:500f****8.**P. Erdös and P. Turán, On the distribution of roots of polynomials,*Ann. Math.*51:105-119, 1950. MR**11:431b****9.**C. G. Esseen,*On the concentration function of a sum of independent random variables*, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete**9**(1968), 290–308. MR**231419**, https://doi.org/10.1007/BF00531753**10.**J. M. Hammersley, The zeroes of a random polynomial,*Proc. Third Berkeley Symposium on Probability and Statistics*, Vol. II:89-111, 1956. MR**18:941c****11.**I. A. Ibragimov and Yu. V. Linnik,*Independent and stationary sequences of random variables*, Wolters-Noordhoff Publishing, Groningen, 1971. With a supplementary chapter by I. A. Ibragimov and V. V. Petrov; Translation from the Russian edited by J. F. C. Kingman. MR**0322926****12.**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****13.**M. Kac, On the average number of real roots of a random algebraic equation.*Bull. Amer. Math. Society*18:29-35, 1943.49: 314-320, 938, 1943. MR**4:196d;****5**:1792**14.**Mark Kac,*Probability and related topics in physical sciences*, With special lectures by G. E. Uhlenbeck, A. R. Hibbs, and B. van der Pol. Lectures in Applied Mathematics. Proceedings of the Summer Seminar, Boulder, Colo., vol. 1957, Interscience Publishers, London-New York, 1959. MR**0102849****15.**B. Ja. Levin,*Distribution of zeros of entire functions*, American Mathematical Society, Providence, R.I., 1964. MR**0156975****16.**J. Littlewood and A. Offord, On the number of real roots of a random algebraic equation,*J. London Math. Soc.*, 13:288-295, 1938.**17.**S. O. Rice, Mathematical analysis of random noise.*Bell System Technical Journal*, 25:46-156, 1945. MR**6:233i****18.**Larry A. Shepp and Robert J. Vanderbei,*The complex zeros of random polynomials*, Trans. Amer. Math. Soc.**347**(1995), no. 11, 4365–4384. MR**1308023**, https://doi.org/10.1090/S0002-9947-1995-1308023-8

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (1991):
34F05.,
26C10,
30B20

Retrieve articles in all journals with MSC (1991): 34F05., 26C10, 30B20

Additional Information

**Ildar Ibragimov**

Affiliation:
Mathematics Institute, Fontanka 27, St. Petersburg 191011, Russia

Email:
ibr32@pdmi.ras.ru

**Ofer Zeitouni**

Affiliation:
Department of Electrical Engineering, Technion-Israel Institute of Technology, Haifa 32000, Israel

Email:
zeitouni@ee.technion.ac.il

DOI:
https://doi.org/10.1090/S0002-9947-97-01766-2

Keywords:
Random polynomials,
complex roots,
domain of attraction of the stable law

Received by editor(s):
December 2, 1995

Additional Notes:
The work of the first author was partially supported by the Russian Foundation for Fundamental Research, grant 94-01-00301, and by grants R36000 and R36300 of the International Scientific Foundation.

The work of the second author was done while he visited MIT, under support from NSF grant 9302709–DMS

Article copyright:
© Copyright 1997
American Mathematical Society