Real zeroes of random polynomials, I: Flip-invariance, Turán's lemma, and the Newton-Hadamard polygon

2016/01/19 by Ken Söze, Söze, Ken
Mathematics · #26C10 #60-XX #Complex Variables (math.CV) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Point processes and geometric inequalities #Probability (math.PR) #math.CV #math.PR #msc:26C10 #msc:60-XX

paper · pdf · doi:10.48550/arxiv.1601.04850

16 pages

arxiv created 2016/01/19 · openalex publication_date 2016/01/19 · arxiv updated 2016/01/20 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

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.

Citations

• Expected Number of Real Zeros of a Random Polynomial with Independent Identically Distributed Symmetric Long-Tailed Coefficients

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