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Zeros of polynomials with random coefficients. (English) Zbl 1309.26016
The authors study the distribution of zeros of collections of polynomials \(P_n(z) = \sum_{k=0}^n A_k z^k\) with random coefficients. Under mild conditions on the probability distribution of the coefficients, the zeros are asymptotically equidistributed near the unit circle \(\mathbb T = \{z \in \mathbb C : |z|=1\}\). More precisely, the zero counting measure \(\tau_n := \frac 1 n \sum_{k=1}^n \delta_{Z_k}\), where \(Z_1, \ldots,Z_n\) are the zeros of \(P_n\), converges in the weak* topology to the normalized arc-length measure \(\mu_{\mathbb T}\) almost surely. In the paper under review, the authors provide quantitative estimates for the discrepancy of the measures \(\tau_n\) and \(\mu_{\mathbb T}\) in annular sectors and for the expected number of zeros in certain sets. Also random polynomials spanned by general bases are considered. The paper generalizes the results of I. E. Pritsker and A. A. Sola [Proc. Am. Math. Soc. 142, No. 12, 4251–4263 (2014; Zbl 1301.30006)], in particular, several unnecessary restrictions, e.g., the assumption that the coefficients are independent and identically distributed, are removed.

26C10 Real polynomials: location of zeros
30C10 Polynomials and rational functions of one complex variable
60B10 Convergence of probability measures
60B20 Random matrices (probabilistic aspects)
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