Ken Söze | Searcharxiv

arXiv2016

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

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.

math.PR↗

arXiv2016

Real zeroes of random polynomials, II: Descartes' rule of signs and anti-concentration on the symmetric group

In this sequel to Part-I, we present a different approach to bounding the expected number of real zeroes of random polynomials with real independent identically distributed coefficients or more generally, exchangeable coefficients. We show that the mean number of real zeroes does not grow faster than the logarithm of the degree. The main ingredients of our approach are Descartes' rule of signs and a new anti-concentration inequality for the symmetric group. This paper can be read independently of part-I in this series.

math.PR↗