Abstract
We show that with high probability the number of real zeroes of a random polynomial is bounded by the number of vertices on its Newton–Hadamard polygon times the cube of the logarithm of the polynomial degree. A similar estimate holds for zeroes lying on any curve in the complex plane, which is the graph of a Lipschitz function in polar coordinates. The proof is based on the classical Turán lemma.
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To Ildar Ibragimov with admiration
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Söze, K. Real zeroes of random polynomials, I. Flip-invariance, Turán’s lemma, and the Newton-Hadamard polygon. Isr. J. Math. 220, 817–836 (2017). https://doi.org/10.1007/s11856-017-1535-6
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DOI: https://doi.org/10.1007/s11856-017-1535-6