On Geometry of the Zeros of a Polynomial

Let p ( z ) = a 0 + a 1 z + a 2 z 2 + a 3 z 3 + ⋯ + a n z n $$p(z) = a_{0} + a_{1}z + a_{2}z^{2} + a_{3}z^{3} + \cdots + a_{n}z^{n}$$ be a polynomial of degree n, where the coefficients a k may be complex. The problem of locating the zeros of a polynomial p(z) is a long-standing classical problem which has frequently been investigated. These problems, besides being of theoretical interest, have important applications in many scientific specialization areas, such as coding theory, cryptography, combinatorics, number theory, mathematical biology, engineering, signal processing, communication theory, and control theory, and for this reason there is always a need for better and sharper results. This paper is expository in nature, and here we make an attempt to provide a systematic study of these problems by presenting some results starting from the results of Gauss and Cauchy, who we believe were the earliest contributors in this subject, to some of the most recent ones. When possible, we have tried to present the proofs of some of the theorems. Also, included here are some results on evaluating the quality of bounds by using numerical methods or MATLAB.