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On Geometry of the Zeros of a Polynomial

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Computation, Cryptography, and Network Security

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Abstract

Let \(p(z) = a_{0} + a_{1}z + a_{2}z^{2} + a_{3}z^{3} + \cdots + a_{n}z^{n}\) 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.

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Notes

  1. 1.

    Theorem 3.6 was proved independently and by a different method by Mishael Zedek [72].

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Auburn University, Auburn, AL, 36849, USA

    N. K. Govil & Eze R. Nwaeze

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  1. N. K. Govil
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Correspondence to N. K. Govil .

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  1. Department of Mathematics and Engineering, Hellenic Military Academy, Vari Attikis, Greece

    Nicholas J. Daras

  2. Department of Mathematics, ETH Zürich, Zürich, Switzerland

    Michael Th. Rassias

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Govil, N.K., Nwaeze, E.R. (2015). On Geometry of the Zeros of a Polynomial. In: Daras, N., Rassias, M. (eds) Computation, Cryptography, and Network Security. Springer, Cham. https://doi.org/10.1007/978-3-319-18275-9_10

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