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Pairing of zeros and critical points for random polynomials. (English. French summary) Zbl 1373.30006

Summary: Let \(p_N\) be a random degree \(N\) polynomial in one complex variable whose zeros are chosen independently from a fixed probability measure \(\mu\) on the Riemann sphere \(S^{2}\). This article proves that if we condition \(p_{N}\) to have a zero at some fixed point \(\xi\in S^{2}\), then, with high probability, there will be a critical point \(w_\xi\) at a distance \(N^{-1}\) away from \(\xi\). This \(N^{-1}\) distance is much smaller than the \(N^{-1/2}\) typical spacing between nearest neighbors for \(N\) i.i.d. points on \(S^{2}\). Moreover, with the same high probability, the argument of \(w_\xi\) relative to \(\xi\) is a deterministic function of \(\mu\) plus fluctuations on the order of \(N^{-1}\).

MSC:

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)
60G60 Random fields
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