Article
Relationship between the zeros of two polynomials
Department of Mathematics, The University of Hong Kong, Pokfulam, Hong Kong
Linear Algebra and its Applications 01/2010; DOI: 10.1016/j.laa.2009.07.028  Citations (12)
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 Mathematische Annalen 01/1909; 66(4):488510. · 1.38 Impact Factor

Article: An Analog of the Poincarée Separation Theorem for Normal Matrices and the Gauss–Lucas Theorem
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ABSTRACT: We establish an analog of the Cauchy–Poincare separation theorem for normal matrices in terms of majorization. A solution to the inverse spectral problem (Borg type result) is also presented. Using this result, we generalize and extend the Gauss–Lucas theorem about the location of roots of a complex polynomial and of its derivative. The generalization is applied to prove old conjectures due to de Bruijn–Springer and Schoenberg.Functional Analysis and Its Applications 06/2003; 37(3):232235. · 0.53 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: In this paper, we introduce a new type of companion matrices, namely, Dcompanion matrices. By using these Dcompanion matrices, we are able to apply matrix theory directly to study the geometrical relation between the zeros and critical points of a polynomial. In fact, this new approach will allow us to prove quite a number of new as well as known results on this topic. For example, we prove some results on the majorization of the critical points of a polynomial by its zeros. In particular, we give a different proof of a recent result of Gerhard Schmeisser on this topic. The same method allows us to prove a higher order Schoenbergtype conjecture proposed by M.G. de Bruin and A. Sharma.Journal of Mathematical Analysis and Applications 01/2006; · 1.05 Impact Factor
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