Article

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

ABSTRACT

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
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    • "Later, in [8] they generalized this approach by one-rank perturbation technique. Note that in [8], W. Cheung and T. Ng partially rediscover the so-called " one-rank perturbation method " developed by Yu. Barkovsky in his PhD thesis [1] which, unfortunately, was only partially published. "
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    • "Q(z) = D ξ P (z) = nP (z) − (z − ξ)P (z) = nξ − n j=1 z j z n−1 + · · · . · a Sz.-Nagy's generalized derivative of P if n j=1 λ j = n and λ j > 0 ([5] "
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    ABSTRACT: For random polynomials with i.i.d. (independent and identically distribu-ted) zeros following any common probability distribution $\mu$ with support contained in the unit circle, the empirical measures of the zeros of their first and higher order derivatives will be proved to converge weakly to $\mu$ a.s. (almost sure(ly)). This, in particular, completes a recent work of Subramanian on the first order derivative case where $\mu$ was assumed to be non-uniform. The same a.s. weak convergence will also be shown for polar and Sz.-Nagy's generalized derivatives, on some mild conditions.
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    ABSTRACT: For random polynomials with independent and identically distributed (i.i.d.) zeros following any common probability distribution \(\mu \) with support contained in the unit circle, the empirical measures of the zeros of their first and higher-order derivatives will be proved to converge weakly to \(\mu \) almost surely (a.s.). This, in particular, completes a recent work of Subramanian on the first-order derivative case where \(\mu \) was assumed to be non-uniform. The same almost sure weak convergence will also be shown for polar and Sz.-Nagy’s generalized derivatives, assuming some mild conditions.
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