On the location of critical points of polynomials
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- by Branko Ćurgus and Vania Mascioni PDF
- Proc. Amer. Math. Soc. 131 (2003), 253-264 Request permission
Abstract:
Given a polynomial $p$ 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 $\omega (p,\alpha ) := \min \bigl \{|\alpha - v| : v \in Z(p)\setminus \{\alpha \} \bigr \}$ and $\tau (p,\alpha ) := \min \bigl \{|\alpha - v| : v \in Z(p’)\setminus \{\alpha \} \bigr \}$. 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 $Z(p)$. 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 $p$. 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 $n$.References
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Additional Information
- Branko Ćurgus
- Affiliation: Department of Mathematics, Western Washington University, Bellingham, Washington 98225
- Email: curgus@cc.wwu.edu
- Vania Mascioni
- Affiliation: Department of Mathematics, Western Washington University, Bellingham, Washington 98225
- Email: masciov@cc.wwu.edu
- Received by editor(s): July 10, 2001
- Received by editor(s) in revised form: September 4, 2001
- Published electronically: June 3, 2002
- Communicated by: N. Tomczak-Jaegermann
- © Copyright 2002 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 131 (2003), 253-264
- MSC (2000): Primary 30C15; Secondary 26C10
- DOI: https://doi.org/10.1090/S0002-9939-02-06534-6
- MathSciNet review: 1929045