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, \[ {P_n}(z) = \sum \limits _{j = 0}^{n - 1} {{\eta _j}{z^j}} ,\] 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 \[ {\mathbf {E}}{\nu _n}(\Omega ) = \int _\Omega {{g_n}(x) dx.} \] 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 \[ {\mathbf {E}}{\nu _n}(\Omega ) = \int _\Omega {{h_n}(x,y) dxdy + \int _{\Omega \cap \mathbb {R}} {{g_n}(x) dx,} } \] 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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© Copyright 1995
American Mathematical Society