On the expected number of zeros of a random harmonic polynomial

Li, Wenbo; Wei, Ang

doi:10.1090/S0002-9939-08-09555-5

On the expected number of zeros of a random harmonic polynomial
HTML articles powered by AMS MathViewer

by Wenbo V. Li and Ang Wei PDF

Proc. Amer. Math. Soc. 137 (2009), 195-204 Request permission

Abstract:

We study the distribution of complex zeros of Gaussian harmonic polynomials with independent complex coefficients. The expected number of zeros is evaluated by applying a formula of independent interest for the expected absolute value of quadratic forms of Gaussian random variables.

References

Jean-Marc Azaïs and Mario Wschebor, On the distribution of the maximum of a Gaussian field with $d$ parameters, Ann. Appl. Probab. 15 (2005), no. 1A, 254–278. MR 2115043, DOI 10.1214/105051604000000602
J.-M. Azaïs and M. Wschebor, A self contained proof of the Rice formula for random fields, (2006), preprint available at: http://www.lsp.ups-tsle.fr/ Azais/publi/completeproof.pdf.
Pavel Bleher and Xiaojun Di, Correlations between zeros of a random polynomial, J. Statist. Phys. 88 (1997), no. 1-2, 269–305. MR 1468385, DOI 10.1007/BF02508472
Daoud Bshouty, Walter Hengartner, and Tiferet Suez, The exact bound on the number of zeros of harmonic polynomials, J. Anal. Math. 67 (1995), 207–218. MR 1383494, DOI 10.1007/BF02787790
Daoud Bshouty and Abdallah Lyzzaik, On Crofoot-Sarason’s conjecture for harmonic polynomials, Comput. Methods Funct. Theory 4 (2004), no. 1, 35–41. MR 2081663, DOI 10.1007/BF03321053
Felipe Cucker and Mario Wschebor, On the expected condition number of linear programming problems, Numer. Math. 94 (2003), no. 3, 419–478. MR 1981163, DOI 10.1007/s00211-002-0385-1
J. E. A. Dunnage, The number of real zeros of a random trigonometric polynomial, Proc. London Math. Soc. (3) 16 (1966), 53–84. MR 192532, DOI 10.1112/plms/s3-16.1.53
Peter Duren, Walter Hengartner, and Richard S. Laugesen, The argument principle for harmonic functions, Amer. Math. Monthly 103 (1996), no. 5, 411–415. MR 1400723, DOI 10.2307/2974933
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, DOI 10.1090/S0273-0979-1995-00571-9
Lukas Geyer, Sharp bounds for the valence of certain harmonic polynomials, Proc. Amer. Math. Soc. 136 (2008), no. 2, 549–555. MR 2358495, DOI 10.1090/S0002-9939-07-08946-0
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
M. Kac, On the average number of real roots of a random algebraic equation, Bull. Amer. Math. Soc. 49 (1943), 314–320. MR 7812, DOI 10.1090/S0002-9904-1943-07912-8
Dmitry Khavinson and Genevra Neumann, On the number of zeros of certain rational harmonic functions, Proc. Amer. Math. Soc. 134 (2006), no. 4, 1077–1085. MR 2196041, DOI 10.1090/S0002-9939-05-08058-5
Dmitry Khavinson and Grzegorz Świa̧tek, On the number of zeros of certain harmonic polynomials, Proc. Amer. Math. Soc. 131 (2003), no. 2, 409–414. MR 1933331, DOI 10.1090/S0002-9939-02-06476-6
Eric Kostlan, On the expected number of real roots of a system of random polynomial equations, Foundations of computational mathematics (Hong Kong, 2000) World Sci. Publ., River Edge, NJ, 2002, pp. 149–188. MR 2021981
W.V. Li and A. Wei, Gaussian integrals for absolute value functions, in preparation.
Yuval Peres and Bálint Virág, Zeros of the i.i.d. Gaussian power series: a conformally invariant determinantal process, Acta Math. 194 (2005), no. 1, 1–35. MR 2231337, DOI 10.1007/BF02392515
T. Sheil-Small, Complex polynomials, Cambridge Studies in Advanced Mathematics, vol. 75, Cambridge University Press, Cambridge, 2002. MR 1962935, DOI 10.1017/CBO9780511543074
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, DOI 10.1090/S0002-9947-1995-1308023-8
Y. L. Tong, The multivariate normal distribution, Springer Series in Statistics, Springer-Verlag, New York, 1990. MR 1029032, DOI 10.1007/978-1-4613-9655-0
A. S. Wilmshurst, The valence of harmonic polynomials, Proc. Amer. Math. Soc. 126 (1998), no. 7, 2077–2081. MR 1443416, DOI 10.1090/S0002-9939-98-04315-9