On roots of random polynomials
Authors:
Ildar Ibragimov and Ofer Zeitouni
Journal:
Trans. Amer. Math. Soc. 349 (1997), 24272441
MSC (1991):
Primary 34F05.; Secondary 26C10, 30B20
MathSciNet review:
1390040
Fulltext 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 (82h:60103)
 2.
Ludwig
Arnold, Über die Nullstellenverteilung zufälliger
Polynome, Math. Z. 92 (1966), 12–18 (German).
MR
0200966 (34 #851)
 3.
A. T. BharuchaReid 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 YorkLondonSydney, 1976. Wiley Series in
Probability and Mathematical Statistics. MR 0436272
(55 #9219)
 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
(91m:60002)
 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 (95m:60082), http://dx.doi.org/10.1090/S027309791995005719
 7.
Paul
Erdös and A.
C. Offord, On the number of real roots of a random algebraic
equation, Proc. London Math. Soc. (3) 6 (1956),
139–160. MR 0073870
(17,500f)
 8.
P.
Erdös and P.
Turán, On the distribution of roots of polynomials,
Ann. of Math. (2) 51 (1950), 105–119. MR 0033372
(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 0231419
(37 #6974)
 10.
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
(18,941c)
 11.
I.
A. Ibragimov and Yu.
V. Linnik, Independent and stationary sequences of random
variables, WoltersNoordhoff 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
(48 #1287)
 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 (44 #3371)
 13.
M. Kac, On the average number of real roots of a random algebraic equation. Bull. Amer. Math. Society 18:2935, 1943.49: 314320, 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, LondonNew York,
1959. MR
0102849 (21 #1635)
 15.
B.
Ja. Levin, Distribution of zeros of entire functions, American
Mathematical Society, Providence, R.I., 1964. MR 0156975
(28 #217)
 16.
J. Littlewood and A. Offord, On the number of real roots of a random algebraic equation, J. London Math. Soc., 13:288295, 1938.
 17.
S.
O. Rice, Mathematical analysis of random noise, Bell System
Tech. J. 24 (1945), 46–156. MR 0011918
(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
(96a:30006), http://dx.doi.org/10.1090/S00029947199513080238
 1.
 R. J. Adler, The geometry of random fields. Wiley, New York, 1981. MR 82h:60103
 2.
 L. Arnold, Über die Nullstellung verteilung zufälligen Polynome, Math Z., 92:1218, 1966. MR 34:851
 3.
 A. T. BharuchaReid and M. Sambandham, Random polynomials, Academic Press, New York, 1986. MR 87:60118
 4.
 R. N. Bhattacharya and R. Ranga Rao, Normal approximations and asymptotic expansions, Wiley, New York, 1976. MR 55:9219
 5.
 R. Durrett, Probability: Theory and Examples, Wadsworth, Pacific Grove, CA, 1991. MR 91m:60002
 6.
 A. Edelman and E. Kostlan, How many zeros of a random polynomial are real? Bull. of the AMS 32:137, 1995. MR 95m:60082
 7.
 P. Erdös and A. C. Offord, On the number of real roots of a random algebraic equation, Proc. London Math. Soc. 3:139160, 1956. MR 17:500f
 8.
 P. Erdös and P. Turán, On the distribution of roots of polynomials, Ann. Math. 51:105119, 1950. MR 11:431b
 9.
 C. G. Esseen, On the concentration function of a sum of independent random variables, Z. Wahrsch. Verw. Gebiete 9:290308, 1968. MR 37:6974
 10.
 J. M. Hammersley, The zeroes of a random polynomial, Proc. Third Berkeley Symposium on Probability and Statistics, Vol. II:89111, 1956. MR 18:941c
 11.
 I. A. Ibragimov and Y. V. Linnik, Independent and stationary sequences of random variables, WoltersNoordhoff, Groningen, 1971. MR 48:1287
 12.
 I. A. Ibragimov and N. B. Maslova, On the expected number of real zeros of random polynomials. I. Coefficients with zero means. Teor. Veroyatn. Primen., 16:229248, 1971; English transl., Theory Probab. Appl. 16: 228248, 1971. MR 44:3371
 13.
 M. Kac, On the average number of real roots of a random algebraic equation. Bull. Amer. Math. Society 18:2935, 1943.49: 314320, 938, 1943. MR 4:196d; 5:1792
 14.
 M. Kac, Probability and related topics in physical sciences. Interscience, London, 1959. MR 21:1635
 15.
 B. Ja. Levin, Distribution of zeros of entire functions, AMS, Providence 1964. MR 28:217
 16.
 J. Littlewood and A. Offord, On the number of real roots of a random algebraic equation, J. London Math. Soc., 13:288295, 1938.
 17.
 S. O. Rice, Mathematical analysis of random noise. Bell System Technical Journal, 25:46156, 1945. MR 6:233i
 18.
 L. Shepp and R. J. Vanderbei, The complex zeros of random polynomials. Trans. AMS, 347:43654383, 1995. MR 96a:30006
Similar Articles
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, TechnionIsrael Institute of Technology, Haifa 32000, Israel
Email:
zeitouni@ee.technion.ac.il
DOI:
http://dx.doi.org/10.1090/S0002994797017662
PII:
S 00029947(97)017662
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 940100301, 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
