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Transactions of the American Mathematical Society

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The complex zeros of random polynomials


Authors: Larry A. Shepp and Robert J. Vanderbei
Journal: Trans. Amer. Math. Soc. 347 (1995), 4365-4384
MSC: Primary 30C15; Secondary 60G99
DOI: https://doi.org/10.1090/S0002-9947-1995-1308023-8
MathSciNet review: 1308023
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Abstract: Mark Kac gave an explicit formula for the expectation of the number, $ {\nu _n}(\Omega )$, of zeros of a random polynomial,

$\displaystyle {P_n}(z) = \sum\limits_{j = 0}^{n - 1} {{\eta _j}{z^j}} ,$

in any measurable subset $ \Omega $ of the reals. Here, $ {\eta _0}, \ldots ,{\eta _{n - 1}}$ are independent standard normal random variables. In fact, for each $ n > 1$, he obtained an explicit intensity function $ {g_n}$ for which

$\displaystyle {\mathbf{E}}{\nu _n}(\Omega ) = \int_\Omega {{g_n}(x)\,dx.} $

Here, we extend this formula to obtain an explicit formula for the expected number of zeros in any measurable subset $ \Omega $ of the complex plane $ \mathbb{C}$. Namely, we show that

$\displaystyle {\mathbf{E}}{\nu _n}(\Omega ) = \int_\Omega {{h_n}(x,y)\,dxdy + \int_{\Omega \cap \mathbb{R}} {{g_n}(x)\,dx,} } $

where $ {h_n}$ is an explicit intensity function. We also study the asymptotics of $ {h_n}$ showing that for large $ n$ its mass lies close to, and is uniformly distributed around, the unit circle.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1995-1308023-8
Article copyright: © Copyright 1995 American Mathematical Society

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