Zero Distribution of Random Bernoulli Polynomial Mappings

TL;DR

This paper investigates the asymptotic distribution of zeros in multivariable random Bernoulli polynomial systems, showing that their zeros become uniformly distributed on the unit torus as the degree increases.

Contribution

It establishes the asymptotic zero distribution for multivariable Bernoulli polynomial systems, demonstrating convergence to the Haar measure on the unit torus.

Findings

Zeros are discrete with high probability

Empirical zero measures converge to Haar measure

Zeros distribute uniformly on the unit torus

Abstract

In this note, we study asymptotic zero distribution of multivariable full system of random polynomials with independent Bernoulli coefficients. We prove that with overwhelming probability their simultaneous zeros sets are discrete and the associated normalized empirical measure of zeros asymptotic to the Haar measure on the unit torus.