On roots of random polynomials. (English) Zbl 0872.30002
Summary: We study the distribution of the complex roots of random polynomials of degree \(n\) 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.
MSC:
| 30B20 | Random power series in one complex variable |
| 26C10 | Real polynomials: location of zeros |
| 34F05 | Ordinary differential equations and systems with randomness |
References:
| [1] | Robert J. Adler, The geometry of random fields, John Wiley & Sons, Ltd., Chichester, 1981. Wiley Series in Probability and Mathematical Statistics. · Zbl 0478.60059 |
| [2] | Ludwig Arnold, Über die Nullstellenverteilung zufälliger Polynome, Math. Z. 92 (1966), 12 – 18 (German). · Zbl 0139.35801 · doi:10.1007/BF01140538 |
| [3] | A. T. Bharucha-Reid and M. Sambandham, Random polynomials, Academic Press, New York, 1986. · Zbl 0615.60058 |
| [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. · Zbl 0331.41023 |
| [5] | Richard Durrett, Probability, The Wadsworth & Brooks/Cole Statistics/Probability Series, Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA, 1991. Theory and examples. · Zbl 0709.60002 |
| [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. · Zbl 0820.34038 |
| [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. · Zbl 0070.01702 |
| [8] | P. Erdös and P. Turán, On the distribution of roots of polynomials, Ann. Math. 51:105-119, 1950. |
| [9] | C. G. Esseen, On the concentration function of a sum of independent random variables, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 9 (1968), 290 – 308. · Zbl 0195.19303 · doi: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. |
| [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. · Zbl 0219.60027 |
| [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). · Zbl 0277.60051 |
| [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. |
| [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. · Zbl 0087.33003 |
| [15] | B. Ja. Levin, Distribution of zeros of entire functions, American Mathematical Society, Providence, R.I., 1964. · Zbl 0152.06703 |
| [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. · Zbl 0020.13604 |
| [17] | S. O. Rice, Mathematical analysis of random noise. Bell System Technical Journal, 25:46-156, 1945. · Zbl 0063.06487 |
| [18] | Larry A. Shepp and Robert J. Vanderbei, The complex zeros of random polynomials, Trans. Amer. Math. Soc. 347 (1995), no. 11, 4365 – 4384. · Zbl 0841.30006 |
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