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A note on the complex roots of complex random polynomials. (English) Zbl 0963.60049
Let \(a_k(\omega)\) and \(b_k(\omega)\), \(\omega\in\Omega\), \(k=0,1, \dots,n\), be sequences of independent, normally distributed random variables defined on a probability space \((\Omega,A,\text{Pr})\) with mean zero and variance one. Denote \(P(z,\omega)\equiv P(z)= \sum^n_{k=0} c_k(\omega)z^k\) where \(c_k(\omega) \equiv c_k=a_k (\omega)+i b_k(\omega)\). For the case of \(b_k\equiv 0\), \(k=0,1,\dots,n\), it is well known that one can find the density function \(h_n (x)\) such that the expected number of real zeros of \(P(x)\) denoted by \(v_n (R)\), \(R\in\mathbb{R}\), satisfies \(v_n(R)= \int_Rh_n(x)dx\), \(R\in\mathbb{R}\). This formula then could be used to obtain an asymptotic formula for the expected number of real zeros. However, in comparison, very little is known about the complex roots of \(P(z)\). It is obvious, see for example J. E. A. Dunnage [Proc. Lond. Math. Soc., III Ser. 18, 439-460 (1968; Zbl 0164.19002)], that there could be no analogue of the above asymptotic formula for the expected number of real zeros for \(P(z)\) with complex coefficients. Also the problems concerning the complex roots are involved with analysis of the different levels of difficulties. This can be seen in the classical work of J. M. Hammersley [in: Proc. 3rd Berkeley Sympos. Math. Statist. Probab. 2, 89-111 (1956; Zbl 0074.34302)] who obtained \(h_n(z)\) where the coefficients of \(P(z)\) are multivariate normal random vector with mean \(\mu_c\) and a nondegenerate covariance matrix.
However, recently I. Ibragimov and O. Zeitouni [Trans. Am. Math. Soc. 349, No. 6, 2427-2441 (1997; Zbl 0872.30002)] used an idea of R. J. Adler [“The geometry of random fields” (1981; Zbl 0478.60059)] and significantly simplified the derivation of the density of complex roots. A complete and comprehensive background of up-to-date results can be found in a book by this reviewer [“Topics on random polynomials” (1998; Zbl 0949.60010)], which also includes the pioneer work of L. A. Shepp and R. J. Vanderbei [Trans. Am. Math. Soc. 347, No. 11, 4365-4384 (1995; Zbl 0841.30006)]. The present paper makes an interesting application of the new developments in studying the complex roots of \(P(z)\). It not only shows that the above results due to Hammersley can be obtained easily, but they can also be generalized further. The generalization includes the general assumptions for the cofficients \(c\) and the case when \(P(z)=x+iy\), \(x,y\in\mathbb{R}\), which, as far as the multivariate normal is concerned, is the most general case studied to date.

MSC:
60G99 Stochastic processes
30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
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[1] Adler, R.J., 1981. The Geometry of Random Fields. Wiley, New York. · Zbl 0478.60059
[2] Bharucha-Reid, A.T., Sambandham, M., 1986. Random Polynomials. Academic Press, New York. · Zbl 0615.60058
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