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Sums of two polynomials with each having real zeros symmetric with the other

  • Seon-Hong Kim
Article

Abstract

Consider the polynomial equation
$$\prod\limits_{i = 1}^n {(x - r_i )} + \prod\limits_{i = 1}^n {(x + r_i )} = 0,$$
where 0 <r 1 ⪯ {irt}2⪯... ⪯r n All zeros of this equation lie on the imaginary axis. In this paper, we show that no two of the zeros can be equal and the gaps between the zeros in the upper half-plane strictly increase as one proceeds upward. Also we give some examples of geometric progressions of the zeros in the upper half-plane in casesn = 6, 8, 10.

Keywords

Polynomial zero geometric progression 

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References

  1. [1]
    Fell H J, On the zeros of convex combinations of polynomials,Pacific J. Math. 89 (1980) 43–50MATHMathSciNetGoogle Scholar
  2. [2]
    Marden M, Geometry of Polynomials,Math. Surveys, No. 3, Amer. Math. Society, Providence, R.I., 1966.Google Scholar

Copyright information

© Indian Academy of Sciences 2002

Authors and Affiliations

  1. 1.School of Mathematical SciencesSeoul National UniversitySeoulKorea

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