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

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 Tur\'an lemma.