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Outline

Title
Abstract
FAQs
Introduction
The Main Result
Conclusion
References
All Topics
Mathematics
Algebra

Note on the location of zeros of polynomials

Profile image of Josep Rubio-masseguJosep Rubio-massegu

2011

Abstract

In this note, we provide a wide range of upper bounds for the moduli of the zeros of a complex polynomial. The obtained bounds complete a series of previous papers on the location of zeros of polynomials.

FAQs

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What are the key improvements in bounds for polynomial zeros in this study?add

Theorem 3 establishes tighter bounds for polynomial zeros, yielding |z| < 1 + εℓ, refining previous results from Theorem 2.

How does Theorem 2 compare with Theorem 1 in terms of Zeros' bounds?add

For ℓ = 2, Theorem 1 provides a better bound than Theorem 2; however, Theorem 3 incorporates both, enhancing overall accuracy.

What practical applications derive from the results on polynomial zeros?add

The location of polynomial zeros is crucial in fields like control systems, signal processing, and electrical networks, enhancing computational efficiency.

What methodologies led to the formulation of Theorem 3's results?add

The authors utilized a comparative analysis of coefficient magnitudes and derived upper bounds via polynomial properties and root behavior.

When is the best time to use Theorem 8 instead of Theorem 3?add

Theorem 8 simplifies the auxiliary equations involved, making it preferable when clarity is needed in computing the zeros' bounds.

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References (8)

  1. M. Marden, Geometry of Polynomials, American Mathematical Society, Providence, Rhode Island, 1966.
  2. D. Mitrinovic, G. Milovanovic, T. M. Rassias, Topics in Polynomials: Extremal Problems, Inequalities and Zeros, World Scientific, Singapore, 1994.
  3. Q. I. Rahman, G. Schmeisser, Analytic Theory of Polynomials, Clarendon Press, Oxford, 2002.
  4. A. L. Cauchy, Exercises de mathématique, IV Annee de Bure Freres, Paris, 1829.
  5. A. Joyal, G. Labelle, Q.I. Rahman, On the location of zeros of polynomials, Canad. Math. Bull. 10 (1967) 53-63.
  6. C. Affane-Aji, N. Agarwal, N. K. Govil, Location of zeros of polynomials, Math. Comput. Modelling 50 (2009) 306-313.
  7. Y. J. Sun, J. G. Hsieh, A note on the circular bound of polynomial zeros, IEEE Trans. Circuits Syst. I 43 (1996) 476-478.
  8. V.K. Jain, On Cauchy's bound for zeros of a polynomial, Turkish J. Math. 30 (2006) 95-100.

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