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  On Multiple Roots in Descartes' Rule and Their Distance to Roots of Higher Derivatives

Eigenwillig, A. (2007). On Multiple Roots in Descartes' Rule and Their Distance to Roots of Higher Derivatives. Journal of Computational and Applied Mathematics, 200(1), 226-230. doi:10.1016/j.cam.2005.12.016.

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Item Permalink: https://hdl.handle.net/11858/00-001M-0000-000F-2019-9 Version Permalink: https://hdl.handle.net/11858/00-001M-0000-0029-BDD3-1
Genre: Journal Article

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Copyright © 2006 Published by Elsevier B.V. This article has been published in Journal of Computational and Applied Mathematics 200(1), March 2007, Pages 226-230.
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 Creators:
Eigenwillig, Arno1, Author           
Affiliations:
1Algorithms and Complexity, MPI for Informatics, Max Planck Society, ou_24019              

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 Abstract: If an open interval $I$ contains a $k$-fold root $\alpha$ of a real polynomial~$f$, then, after transforming $I$ to $(0,\infty)$, Descartes' Rule of Signs counts exactly $k$ roots of $f$ in~$I$, provided $I$ is such that Descartes' Rule counts no roots of the $k$-th derivative of~$f$. We give a simple proof using the Bernstein basis. The above condition on $I$ holds if its width does not exceed the minimum distance $\sigma$ from $\alpha$ to any complex root of the $k$-th derivative. We relate $\sigma$ to the minimum distance $s$ from $\alpha$ to any other complex root of $f$ using Szeg{\H o}'s composition theorem. For integer polynomials, $\log(1/\sigma)$ obeys the same asymptotic worst-case bound as $\log(1/s)$.

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Language(s): eng - English
 Dates: 2008-02-2820072007
 Publication Status: Issued
 Pages: -
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 Table of Contents: -
 Rev. Type: Peer
 Identifiers: eDoc: 356757
DOI: 10.1016/j.cam.2005.12.016
Other: Local-ID: C12573CC004A8E26-630A22C1D279C2D0C125725E0036EB66-Eigenwillig2007a
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Title: Journal of Computational and Applied Mathematics
Source Genre: Journal
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Publ. Info: Antwerpen : North-Holland
Pages: - Volume / Issue: 200 (1) Sequence Number: - Start / End Page: 226 - 230 Identifier: ISSN: 0377-0427
CoNE: https://pure.mpg.de/cone/journals/resource/954926238443