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Random polynomials: central limit theorems for the real roots. (English) Zbl 1482.60041

Problems related to the number of real roots of random polynomials arise in numerous applications. One of the main questions is deriving the limiting distribution for this number. The aim of this paper is to generalise the central limit theorem for the number of real roots of Kac polynomials. The authors consider the polynomials \[ P_n(x)=\sum_{i=1}^n c_i\xi_ix^i, \] where \(\xi_i\) are independent random variables and \(c_i\) are deterministic coefficients.
They suggest a new approach, that allows to derive a general central limit theorem for a large class of random polynomials with coefficients \(c_i\) that grow polynomially. The approach generalises central limit theorems for various previously studies classes of polynomials, including Kac polynomials, their derivatives, hyperbolic polynomials, and others. The authors provide a detailed bibliography and discussion of the known results in the literature.

MSC:

60F05 Central limit and other weak theorems
30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
26C10 Real polynomials: location of zeros
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