Generalization of the Hurwitz Theorem
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Date
2011
Authors
Journal Title
Journal ISSN
Volume Title
Publisher
Suleyman Demirel University
Abstract
This paper presents the main theorem of the author’s diploma work, which provides a generalization of the classical Hurwitz theorem for real-coefficient polynomials. By constructing two auxiliary polynomials and establishing precise relations between their coefficients and the coefficients of the original polynomial, several lemmas are proven to support the main result. The theorem shows that the roots of a polynomial lie in specific half-planes or domains of the complex plane if and only if the associated transformed polynomials are Hurwitz polynomials. This connection extends the classical Hurwitz stability criterion to shifted and reflected regions of the complex plane. The paper also discusses a broader geometric problem involving arbitrary lines in the plane and demonstrates, through counterexamples, that a general solution does not exist for all orientations.
Description
Keywords
Hurwitz polynomial, polynomial roots, complex plane, half-plane conditions, coefficient relations, generalized Hurwitz theorem, transformations of polynomials
Citation
Murzabulatov Meiram/ Generalization of the Hurwitz Theorem / Suleyman Demirel University/ СДУ хабаршысы, 2(18).