TY - JOUR

T1 - The complex zeros of random polynomials

AU - Shepp, Larry

AU - Vanderbei, Robert J.

N1 - Copyright:
Copyright 2016 Elsevier B.V., All rights reserved.

PY - 1995/11

Y1 - 1995/11

N2 - Mark Kac gave an explicit formula for the expectation of the number, vn(Ω), of zeros of a random polynomial, in any measurable subset Ci of the reals. Here, ηo,., ηn-1are independent standard normal random variables. In fact, for each n > 1, he obtained an explicit intensity function g for which Here, we extend this formula to obtain an explicit formula for the expected number of zeros in any measurable subset Ω of the complex plane ℂ. Namely, we show that where hnis an explicit intensity function. We also study the asymptotics of hnshowing that for large n its mass lies close to, and is uniformly distributed around, the unit circle.

AB - Mark Kac gave an explicit formula for the expectation of the number, vn(Ω), of zeros of a random polynomial, in any measurable subset Ci of the reals. Here, ηo,., ηn-1are independent standard normal random variables. In fact, for each n > 1, he obtained an explicit intensity function g for which Here, we extend this formula to obtain an explicit formula for the expected number of zeros in any measurable subset Ω of the complex plane ℂ. Namely, we show that where hnis an explicit intensity function. We also study the asymptotics of hnshowing that for large n its mass lies close to, and is uniformly distributed around, the unit circle.

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U2 - 10.1090/S0002-9947-1995-1308023-8

DO - 10.1090/S0002-9947-1995-1308023-8

M3 - Article

AN - SCOPUS:21844506265

VL - 347

SP - 4365

EP - 4384

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 11

ER -