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On roots of random polynomials
Author(s):
Ildar
Ibragimov;
Ofer
Zeitouni
Journal:
Trans. Amer. Math. Soc.
349
(1997),
2427-2441.
MSC (1991):
Primary 34F05.;
Secondary 26C10, 30B20
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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.
References:
- 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., 92:12-18, 1966. MR 34:851
- 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 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 32:1-37, 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. 3:139-160, 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. Wahrsch. Verw. Gebiete 9:290-308, 1968. MR 37:6974
- 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 Y. V. Linnik, Independent and stationary sequences of random variables, Wolters-Noordhoff, 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., 16:229-248, 1971; English transl., Theory Probab. Appl. 16: 228-248, 1971. MR 44:3371
- 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.
- 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., 13:288-295, 1938.
- 17.
- S. O. Rice, Mathematical analysis of random noise. Bell System Technical Journal, 25:46-156, 1945. MR 6:233i
- 18.
- L. Shepp and R. J. Vanderbei, The complex zeros of random polynomials. Trans. AMS, 347:4365-4383, 1995. MR 96a:30006
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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:
10.1090/S0002-9947-97-01766-2
PII:
S 0002-9947(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
Copyright of article:
Copyright
1997,
American Mathematical Society
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