• Seon-Hong Kim

      Articles written in Proceedings – Mathematical Sciences

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

      Seon-Hong Kim

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      Consider the polynomial equation$$\prod\limits_{i = 1}^n {(x - r_i )} + \prod\limits_{i = 1}^n {(x + r_i )} = 0,$$ where 0 <r1 ⪯ {irt}2⪯... ⪯rn 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.

    • Root and critical point behaviors of certain sums of polynomials

      SEON-HONG KIM SUNG YOON KIM TAE HYUNG KIM SANGHEON LEE

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      It is known that no two roots of the polynomial equation $$\prod^{n}_{j=1}(x-rj)+\prod^{n}_{j=1}(x+rj)=0$$,

      where 0 < $r_{1}\leq r_{2}\leq \cdots\leq r_{n}$, can be equal and the gaps between the roots of (1) in the upper half-plane strictly increase as one proceeds upward, and for 0 < $h$ < $r_{k}$, the roots of $$(x-r_{k}-h)\prod^{n}_{{j=1}\atop{j\neq k}}(x-r_{j})+(x+r_{k}+h)\prod^{n}_{{j=1}\atop{j\neq k}}(x+r_{j})=0$$

      and the roots of (1) in the upper half-plane lie alternatively on the imaginary axis. In this paper, we study how the roots and the critical points of (1) and (2) are located.

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