Equidistribution of zeros of random bernoulli polynomial systems

Official URL: https://risc01.sabanciuniv.edu/record=b3075256_(Table of contents)

In this thesis, we consider the full systems of random polynomials with independent ±1-valued Bernoulli distributed coefficients. In the first part of study, we examine the distribution of common solutions of random Bernoulli systems. In order to determine that whether the common solutions are discrete or not, we focus on the directional resultants of these systems. Using the results obtained from the computations of directional resultants, we prove that common solutions of Bernoulli polynomial systems are discrete outside of an exceptional set En,d which has small probability. Randomizing the deterministic results of D’Andrea, Galligo and Sombra, we prove that outside of En,d, the zeros of Bernoulli polynomial systems are equidistributed towards the Haar measure on the unit torus. In the second part, we focus on the expected zero measures of random Bernoulli systems. We study the angular discrepancies and radius discrepancies of sets of common solutions of random Bernoulli polynomial systems. We prove that the expected angular discrepancy and radius discrepancy approach to zero as the degree of polynomials approaches to infinity. Using these results and appyling the classical method in analysis, we prove that the expected zero measure of Bernoulli polynomial systems converges to Haar measure on the unit disc (S1)n in Cn. Lastly, we generalize these results for the random Bernoulli systems on C2 for more general supports.