Expected discrepancy for zeros of random algebraic polynomials

Pritsker, Igor; Sola, Alan

doi:10.1090/S0002-9939-2014-12147-2

Expected discrepancy for zeros of random algebraic polynomials
HTML articles powered by AMS MathViewer

by Igor E. Pritsker and Alan A. Sola PDF

Proc. Amer. Math. Soc. 142 (2014), 4251-4263 Request permission

Abstract:

We study asymptotic clustering of zeros of random polynomials and show that the expected discrepancy of roots of a polynomial of degree $n$, with not necessarily independent coefficients, decays like $\sqrt {\log n/n}$. Our proofs rely on discrepancy results for deterministic polynomials and on order statistics of a random variable. We also consider the expected number of zeros lying in certain subsets of the plane, such as circles centered on the unit circumference, and polygons inscribed in the unit circumference.

References

Robert J. Adler, The geometry of random fields, Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, Ltd., Chichester, 1981. MR 611857
F. Amoroso and M. Mignotte, On the distribution of the roots of polynomials, Ann. Inst. Fourier (Grenoble) 46 (1996), no. 5, 1275–1291 (English, with English and French summaries). MR 1427125
Vladimir V. Andrievskii and Hans-Peter Blatt, Discrepancy of signed measures and polynomial approximation, Springer Monographs in Mathematics, Springer-Verlag, New York, 2002. MR 1871219, DOI 10.1007/978-1-4757-4999-1
Ludwig Arnold, Über die Nullstellenverteilung zufälliger Polynome, Math. Z. 92 (1966), 12–18 (German). MR 200966, DOI 10.1007/BF01140538
A. T. Bharucha-Reid and M. Sambandham, Random polynomials, Probability and Mathematical Statistics, Academic Press, Inc., Orlando, FL, 1986. MR 856019
Peter Borwein and Tamás Erdélyi, On the zeros of polynomials with restricted coefficients, Illinois J. Math. 41 (1997), no. 4, 667–675. MR 1468873
Peter Borwein, Tamás Erdélyi, and Friedrich Littmann, Polynomials with coefficients from a finite set, Trans. Amer. Math. Soc. 360 (2008), no. 10, 5145–5154. MR 2415068, DOI 10.1090/S0002-9947-08-04605-9
Peter Borwein and Richard Lockhart, The expected $L_p$ norm of random polynomials, Proc. Amer. Math. Soc. 129 (2001), no. 5, 1463–1472. MR 1814174, DOI 10.1090/S0002-9939-00-05690-2
Kwok-Kwong Stephen Choi and Michael J. Mossinghoff, Average Mahler’s measure and $L_p$ norms of unimodular polynomials, Pacific J. Math. 252 (2011), no. 1, 31–50. MR 2862140, DOI 10.2140/pjm.2011.252.31
H. A. David and H. N. Nagaraja, Order statistics, 3rd ed., Wiley Series in Probability and Statistics, Wiley-Interscience [John Wiley & Sons], Hoboken, NJ, 2003. MR 1994955, DOI 10.1002/0471722162
P. Erdös and P. Turán, On the distribution of roots of polynomials, Ann. of Math. (2) 51 (1950), 105–119. MR 33372, DOI 10.2307/1969500
Kambiz Farahmand, Topics in random polynomials, Pitman Research Notes in Mathematics Series, vol. 393, Longman, Harlow, 1998. MR 1679392
G. T. Fielding, The expected value of the integral around the unit circle of a certain class of polynomials, Bull. London Math. Soc. 2 (1970), 301–306. MR 280689, DOI 10.1112/blms/2.3.301
Tord Ganelius, Sequences of analytic functions and their zeros, Ark. Mat. 3 (1954), 1–50. MR 62826, DOI 10.1007/BF02589280
C. P. Hughes and A. Nikeghbali, The zeros of random polynomials cluster uniformly near the unit circle, Compos. Math. 144 (2008), no. 3, 734–746. MR 2422348, DOI 10.1112/S0010437X07003302
Ildar Ibragimov and Ofer Zeitouni, On roots of random polynomials, Trans. Amer. Math. Soc. 349 (1997), no. 6, 2427–2441. MR 1390040, DOI 10.1090/S0002-9947-97-01766-2
Ildar Ibragimov and Dmitry Zaporozhets, On distribution of zeros of random polynomials in complex plane, Prokhorov and contemporary probability theory, Springer Proc. Math. Stat., vol. 33, Springer, Heidelberg, 2013, pp. 303–323. MR 3070481, DOI 10.1007/978-3-642-33549-5_{1}8
Zakhar Kabluchko and Dmitry Zaporozhets, Roots of random polynomials whose coefficients have logarithmic tails, Ann. Probab. 41 (2013), no. 5, 3542–3581. MR 3127891, DOI 10.1214/12-AOP764
Maurice Mignotte, Remarque sur une question relative à des fonctions conjuguées, C. R. Acad. Sci. Paris Sér. I Math. 315 (1992), no. 8, 907–911 (French, with English and French summaries). MR 1187133
Igor E. Pritsker, Means of algebraic numbers in the unit disk, C. R. Math. Acad. Sci. Paris 347 (2009), no. 3-4, 119–122 (English, with English and French summaries). MR 2538096, DOI 10.1016/j.crma.2009.01.002
Igor E. Pritsker, Equidistribution of points via energy, Ark. Mat. 49 (2011), no. 1, 149–173. MR 2784262, DOI 10.1007/s11512-010-0124-2
Efraim Shmerling and Kenneth J. Hochberg, Asymptotic behavior of roots of random polynomial equations, Proc. Amer. Math. Soc. 130 (2002), no. 9, 2761–2770. MR 1900883, DOI 10.1090/S0002-9939-02-06340-2