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

TL;DR

This paper establishes probabilistic bounds on the number of real zeros of random polynomials, linking it to geometric features of the Newton-Hadamard polygon and extending to zeros on certain curves, using Turán's lemma.

Contribution

It introduces a novel probabilistic bound connecting zeros of random polynomials to the Newton-Hadamard polygon's vertices, extending classical results to complex curves.

Findings

Number of real zeros bounded by vertices of Newton-Hadamard polygon times (log degree)^3

Similar bounds for zeros on Lipschitz curves in polar coordinates

Proof utilizes classical Turán's lemma

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.