DOI QR코드

DOI QR Code

ON SOME ROOT BEHAVIORS OF CERTAIN SUMS OF POLYNOMIALS

  • Chong, Han-Kyol (Department of Mathematics Sookmyung Women's University) ;
  • Kim, Seon-Hong (Department of Mathematics Sookmyung Women's University)
  • Received : 2013.12.01
  • Published : 2016.01.31

Abstract

It is known that no two of the roots of the polynomial equation (1) $$\prod\limits_{l=1}^{n}(x-r_l)+\prod\limits_{l=1}^{n}(x+r_l)=0$$, where 0 < $r_1{\leq}r_2{\leq}{\cdots}{\leq}r_n$, can be equal and all of its roots lie on the imaginary axis. In this paper we show that for 0 < h < $r_k$, the roots of $$(x-r_k+h)\prod\limits_{{l=1}\\{l{\neq}k}}^{n}(x-r_l)+(x+r_k-h)\prod\limits_{{l=1}\\{l{\neq}k}}^{n}(x+r_l)=0$$ and the roots of (1) in the upper half-plane lie alternatively on the imaginary axis.

Keywords

sums of polynomials;roots;root squeezing

Acknowledgement

Supported by : Sookmyung Women's University

References

  1. B. Anderson, Polynomial root dragging, Amer. Math. Monthly 100 (1993), no. 9, 864-866. https://doi.org/10.2307/2324665
  2. H. J. Fell, On the zeros of convex combinations of polynomials, Pacific J. Math. 89 (1980), no. 1, 43-50. https://doi.org/10.2140/pjm.1980.89.43
  3. C. Frayer, Squeezing polynomial roots a nonuniform distance, Missouri J. Math. Sci. 22 (2010), no. 2, 124-129.
  4. C. Frayer and J. A. Swenson, Polynomial root motion, Amer. Math. Monthly 117 (2010), no. 7, 641-646. https://doi.org/10.4169/000298910X496778
  5. S.-H. Kim, Sums of two polynomials with each having real zeros symmetric with the other, Proc. Indian Acad. Sci. 112 (2002), no. 2, 283-288.
  6. M.Marden, Geometry of Polynomials, AmericanMathematical Society, Providence, 1966.
  7. Q. I. Rahman and G. Schmeisser, Analytic Theory of Polynomials, Oxford University Press, Oxford, 2002.
  8. T. Sheil-Small, Complex Polynomials, Cambridge University Press, Cambridge, 2002.

Cited by

  1. Root and critical point behaviors of certain sums of polynomials vol.128, pp.2, 2018, https://doi.org/10.1007/s12044-018-0402-7