Probabilistic Approach to Distribution of Zeros and Extremal Problems of Random Polynomials

Abstract

This research aims to fill the gap in the study of random polynomials by intro- ducing and investigating the concept of probabilistic bounds of roots in finite degree polynomials, as well as the probabilistic study of their norms. Contrary to the ample asymptotic studies on high-degree polynomials, the probabilistic bounds for finite degree polynomials are still largely unexplored. Specifically, this work delves into the bounds on the roots of complex random polynomials, studying how these bounds relate to the location of zeros and the feasibility of making probabilistic assertions about their likely regions. It also seeks to examine the probabilities of roots lying within given regions. Furthermore, the probabilistic study of the bounds of muduli of random polynomials on the unit circle has so far received limited at- tention. In addressing these gaps, this research augments the existing knowledge in polynomial theory and probability theory, providing new methodologies applicable to a broad range of problems both within and beyond the theory of random poly- nomials. Consequently, it contributes significantly to the understanding of complex random polynomials and the properties of their roots, with potential implications in mathematics, physics, and engineering. newline