A companion matrix approach to the study of zeros and critical points of a polynomial |
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Authors: | Wai Shun Cheung |
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Institution: | Department of Mathematics, The University of Hong Kong, Pokfulam, Hong Kong |
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Abstract: | In this paper, we introduce a new type of companion matrices, namely, D-companion matrices. By using these D-companion 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 Schoenberg-type conjecture proposed by M.G. de Bruin and A. Sharma. |
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Keywords: | D-companion matrices Polynomials Zeros Critical points Schoenberg conjecture De Bruin and Sharma conjecture Majorization Gerschgorin's disks Ovals of Cassini |
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