Roots of random polynomials with coefficients of polynomial growth. (English) Zbl 1428.60072

Do, Yen; Nguyen, Oanh; Vu, Van

Ann. Probab. 46, No. 5, 2407-2494 (2018).

Summary: In this paper, we prove optimal local universality for roots of random polynomials with arbitrary coefficients of polynomial growth. As an application, we derive, for the first time, sharp estimates for the number of real roots of these polynomials, even when the coefficients are not explicit. Our results also hold for series; in particular, we prove local universality for random hyperbolic series.

MSC:

60G99

Stochastic processes

Keywords:

random polynomials; real roots; complex roots; correlation; arbitrary coefficients; universality

References:

[1]	Alon, N. and Spencer, J. H. (2008). The Probabilistic Method, 3rd ed. Wiley, Hoboken, NJ. · Zbl 1148.05001
[2]	Anderson, G. W., Guionnet, A. and Zeitouni, O. (2010). An Introduction to Random Matrices. Cambridge Studies in Advanced Mathematics118. Cambridge Univ. Press, Cambridge. · Zbl 1184.15023
[3]	Bharucha-Reid, A. T. and Sambandham, M. (1986). Random Polynomials. Academic Press, Orlando, FL. · Zbl 0615.60058
[4]	Bleher, P. and Di, X. (2004). Correlations between zeros of non-Gaussian random polynomials. Int. Math. Res. Not. IMRN2004 2443–2484. · Zbl 1066.60052 · doi:10.1155/S1073792804132418
[5]	Bloch, A. and Pólya, G. (1931). On the Roots of Certain Algebraic Equations. Proc. Lond. Math. Soc. (2) 33 102–114. · JFM 57.0128.03
[6]	Das, M. (1969). The average number of maxima of a random algebraic curve. Proc. Cambridge Philos. Soc.65 741–753. · Zbl 0218.60017 · doi:10.1017/S0305004100003583
[7]	Das, M. (1972). Real zeros of a class of random algebraic polynomials. J. Indian Math. Soc. (N.S.) 36 53–63. · Zbl 0293.60058
[8]	Do, Y., Nguyen, H. and Vu, V. (2015). Real roots of random polynomials: Expectation and repulsion. Proc. Lond. Math. Soc. (3) 111 1231–1260. · Zbl 1356.60084 · doi:10.1112/plms/pdv055
[9]	Dubhashi, D. P. and Panconesi, A. (2009). Concentration of Measure for the Analysis of Randomized Algorithms. Cambridge Univ. Press, Cambridge. · Zbl 1213.60006
[10]	Edelman, A. and Kostlan, E. (1995). How many zeros of a random polynomial are real? Bull. Amer. Math. Soc. (N.S.) 32 1–37. · Zbl 0820.34038 · doi:10.1090/S0273-0979-1995-00571-9
[11]	Erdös, P. and Offord, A. C. (1956). On the number of real roots of a random algebraic equation. Proc. Lond. Math. Soc. (3) 6 139–160. · Zbl 0070.01702
[12]	Farahmand, K. (1998). Topics in Random Polynomials. Pitman Research Notes in Mathematics Series393. Longman, Harlow. · Zbl 0949.60010
[13]	Hough, J. B., Krishnapur, M., Peres, Y. and Virág, B. (2009). Zeros of Gaussian Analytic Functions and Determinantal Point Processes. University Lecture Series51. Amer. Math. Soc., Providence, RI. · Zbl 1190.60038
[14]	Ibragimov, I. A. and Maslova, N. B. (1968). The average number of zeros of random polynomials. Vestnik Leningrad. Univ.23 171–172. · Zbl 0235.60060
[15]	Ibragimov, I. A. and Maslova, N. B. (1971). The average number of real roots of random polynomials. Sov. Math., Dokl.12 1004–1008. · Zbl 0277.60053
[16]	Ibragimov, I. A. and Maslova, N. B. (1971). On the expected number of real zeros of random polynomials. I. Coefficients with zero means. Theory Probab. Appl.16 228–248. · Zbl 0277.60051 · doi:10.1137/1116023
[17]	Ibragimov, I. A. and Maslova, N. B. (1971). On the expected number of real zeros of random polynomials. II. Coefficients with non-zero means. Theory Probab. Appl.16 485–493. · Zbl 0277.60052 · doi:10.1137/1116052
[18]	Kac, M. (1943). On the average number of real roots of a random algebraic equation. Bull. Amer. Math. Soc.49 314–320. · Zbl 0060.28602 · doi:10.1090/S0002-9904-1943-07912-8
[19]	Kac, M. (1949). On the average number of real roots of a random algebraic equation. II. Proc. Lond. Math. Soc. (2) 50 390–408. · Zbl 0033.14702 · doi:10.1112/plms/s2-50.5.390
[20]	Littlewood, J. E. and Offord, A. C. (1943). On the number of real roots of a random algebraic equation. III. Rec. Math. [Mat. Sbornik] N.S.12 277–286. · Zbl 0061.01801
[21]	Littlewood, J. E. and Offord, A. C. (1945). On the distribution of the zeros and \(a\)-values of a random integral function. I. J. Lond. Math. Soc.20 130–136. · Zbl 0060.21902
[22]	Littlewood, J. E. and Offord, A. C. (1948). On the distribution of zeros and \(a\)-values of a random integral function. II. Ann. of Math. (2) 49 885–952. · Zbl 0034.34305 · doi:10.2307/1969404
[23]	Logan, B. F. and Shepp, L. A. (1968). Real zeros of random polynomials. Proc. Lond. Math. Soc. (3) 18 29–35. · Zbl 0245.60047 · doi:10.1112/plms/s3-18.1.29
[24]	Logan, B. F. and Shepp, L. A. (1968). Real zeros of random polynomials. II. Proc. Lond. Math. Soc. (3) 18 308–314. · Zbl 0177.45201 · doi:10.1112/plms/s3-18.2.308
[25]	Maslova, N. B. (1974). On the variance of the number of real roots of random polynomials. Theory Probab. Appl.19 35–52. · Zbl 0321.60007
[26]	Maslova, N. B. (1975). On the distribution of the number of real roots of random polynomials. Theory Probab. Appl.19 461–473. · Zbl 0345.60014
[27]	Nazarov, F., Nishry, A. and Sodin, M. (2014). Log-integrability of Rademacher Fourier series, with applications to random analytic functions. St. Petersburg Math. J.25 467–494. · Zbl 1304.42014 · doi:10.1090/S1061-0022-2014-01300-3
[28]	Nazarov, F. and Sodin, M. (2010). Random complex zeroes and random nodal lines. In Proceedings of the International Congress of Mathematicians, Vol. III 1450–1484. Hindustan Book Agency, New Delhi. · Zbl 1296.30005
[29]	Nazarov, F. and Sodin, M. (2012). Correlation functions for random complex zeroes: Strong clustering and local universality. Comm. Math. Phys.310 75–98. · Zbl 1238.60059 · doi:10.1007/s00220-011-1397-4
[30]	Nguyen, H., Nguyen, O. and Vu, V. (2016). On the number of real roots of random polynomials. Commun. Contemp. Math.18 1550052, 17. · Zbl 1385.60019 · doi:10.1142/S0219199715500522
[31]	Nguyen, H. H. and Vu, V. H. (2013). Small ball probability, inverse theorems, and applications. In Erdös Centennial. Bolyai Soc. Math. Stud.25 409–463. János Bolyai Math. Soc., Budapest. · Zbl 1293.05037
[32]	Peres, Y. and Virág, B. (2005). Zeros of the i.i.d. Gaussian power series: A conformally invariant determinantal process. Acta Math.194 1–35. · Zbl 1099.60037 · doi:10.1007/BF02392515
[33]	Pritsker, I. E. and Varga, R. S. (1999). Weighted rational approximation in the complex plane. J. Math. Pures Appl. (9) 78 177–202. · Zbl 0932.30036
[34]	Rudin, W. (1986). Real and Complex Analysis, 3rd ed. McGraw-Hill, New York. · Zbl 0925.00005
[35]	Sambandham, M. (1979). On the average number of real zeros of a class of random algebraic curves. Pacific J. Math.81 207–215. · Zbl 0411.60066 · doi:10.2140/pjm.1979.81.207
[36]	Sambandham, M., Gore, H. and Farahmand, K. (1998). The average number of point [points] of inflection of random algebraic polynomials. Stoch. Anal. Appl.16 721–731. · Zbl 0920.60039 · doi:10.1080/07362999808809558
[37]	Stevens, D. C. (1969). The average number of real zeros of a random polynomial. Comm. Pure Appl. Math.22 457–477. · Zbl 0167.16604 · doi:10.1002/cpa.3160220403
[38]	Tao, T. and Vu, V. (2015). Local universality of zeroes of random polynomials. Int. Math. Res. Not. IMRN2015 5053–5139. · Zbl 1360.60102 · doi:10.1093/imrn/rnu084
[39]	Tao, T. and Vu, V. (2015). Random matrices: Universality of local spectral statistics of non-Hermitian matrices. Ann. Probab.43 782–874. · Zbl 1316.15042 · doi:10.1214/13-AOP876
[40]	Todhunter, I. (2014). A History of the Mathematical Theory of Probability. Cambridge Univ. Press, Cambridge.
[41]	Wilkins, J. E. Jr. (1988). An asymptotic expansion for the expected number of real zeros of a random polynomial. Proc. Amer. Math. Soc.103 1249–1258. · Zbl 0656.60062 · doi:10.1090/S0002-9939-1988-0955018-1