Relationship between the zeros of two polynomials

Department of Mathematics, The University of Hong Kong, Pokfulam, Hong Kong
Linear Algebra and its Applications (Impact Factor: 0.94). 01/2010; 432(1):107-115. DOI: 10.1016/j.laa.2009.07.028


In this paper, we shall follow a companion matrix approach to study the relationship between zeros of a wide range of pairs of complex polynomials, for example, a polynomial and its polar derivative or Sz.-Nagy’s generalized derivative. We shall introduce some new companion matrices and obtain a generalization of the Weinstein–Aronszajn Formula which will then be used to prove some inequalities similar to Sendov conjecture and Schoenberg conjecture and to study the distribution of equilibrium points of logarithmic potentials for finitely many discrete charges. Our method can also be used to produce, in an easy and systematic way, a lot of identities relating the sums of powers of zeros of a polynomial to that of the other polynomial.

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Available from: Wai-Shun Cheung, Oct 20, 2014