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An exercise on the average number of real zeros of random real polynomials. (English) Zbl 1104.60023

Győri, Ervin (ed.) et al., More sets, graphs and numbers. A salute to Vera Sós and András Hajnal. Berlin: Springer. Budapest: János Bolyai Mathematical Society (ISBN 3-540-32377-5/hbk). Bolyai Society Mathematical Studies 15, 79-92 (2006).
The average number of real zeros of an algebraic polynomial \(P_n (X)=\sum^n_{j=0} a_jX^j\) where the coefficients \(\{a_j\}^n_{j=0}\) are a sequence of random variables is well known. These works include M. Kac [Bull. Am. Math. Soc. 49, 314–320 (1943; Zbl 0060.28602)] who assumed independent and identical standard Gaussian distribution for the coefficients. He found that, for \(n\) large, this average number is asymptotic to \((2/\pi)\log n\). Later it was shown that, in fact, this asymptotic value remains valid for a wider class of distributions belonging to the domain of attractions of normal law. However, there is a significant increase to the average number of real zeros, if we assume a non-identical distribution for the coefficients. That is if the variance of the \(j\)th coefficients become \(n\choose j\) and their means remain zero. In this case A. Edelman and E. Kostlan [Bull Am. Math. Soc. 32, 1–37 (1995; Zbl 0820.34038)], among other results, show that the number of real zeros increases to \(\sqrt n\). The present paper generalizes the above case by assuming that \(\text{var}(a_j)={n\choose j}n^{-\beta j}\), for a given value \(\beta\). For this case in the interval \((a,b)\) the polynomial \(P_n(X)\) has \(\sqrt n(\text{Arctan}(bn^{-\beta/2})-\text{Arctan} (an^{-\beta/2}))\) real zeros. It is also shown that the method used in this paper is applicable to other similar polynomials. The earlier results on this subject are discussed by A. T. Bharacha-Reid and M. Sambandham [“Random Polynomials” (1986; Zbl 0615.60058)] and the reviewer [“Topics in random polynomials” (1998; Zbl 0949.60010)].
For the entire collection see [Zbl 1086.05003].

MSC:

60G99 Stochastic processes
30C10 Polynomials and rational functions of one complex variable
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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