On the location of critical points of polynomials期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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On the location of critical points of polynomials

Authors:	Branko Curgus Vania Mascioni

Institution:	Department of Mathematics, Western Washington University, Bellingham, Washington 98225 ; Department of Mathematics, Western Washington University, Bellingham, Washington 98225

Abstract:	Given a polynomial of degree $n \geq 2$ and with at least two distinct roots let $Z(p) = \{z : p(z) = 0\}$ . For a fixed root $\alpha \in Z(p)$ we define the quantities $\omega(p,\alpha) := \min\bigl\{\vert\alpha - v\vert : v \in Z(p)\setminus \{\alpha\} \bigr\}$ and $\tau(p,\alpha) := \min\bigl\{\vert\alpha - v\vert : v \in Z(p')\setminus \{\alpha\} \bigr\}$ . We also define $\omega(p)$ and $\tau(p)$ to be the corresponding minima of $\omega(p,\alpha)$ and $\tau(p,\alpha)$ as $\alpha$ runs over . Our main results show that the ratios $\tau(p,\alpha)/\omega(p,\alpha)$ and $\tau(p)/\omega(p)$ are bounded above and below by constants that only depend on the degree of . In particular, we prove that $(1/n)\omega(p)\leq\tau(p)\leq\bigl(1/2\sin(\pi/n)\bigr)\omega(p)$ , for any polynomial of degree .

Keywords:	Roots of polynomials critical points of polynomials separation of roots

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