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 Author, Editor(s)
 Author(s): Eigenwillig, Arno dblp
 BibTeX cite key*: Eigenwillig2007a

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

 Journal
 Journal Title*: Journal of Computational and Applied Mathematics Journal's URL: http://www.elsevier.com/locate/cam Download URL for the article: http://www.sciencedirect.com/science?_ob=MImg&_imagekey=B6TYH-4J5C7MG-3-1&_cdi=5619&_user=43521&_orig=browse&_coverDate=03%2F01%2F2007&_sk=997999998&view=c&wchp=dGLbVtz-zSkWA&md5=425b5e336f29c39a4894b0d0ba88ee05&ie=/sdarticle.pdf Language: English

 Publisher
 Publisher's Name: Elsevier Publisher's URL: http://www.elsevier.com Publisher's Address: Amsterdam, The Netherlands ISSN: 0377-0427

 Vol, No, pp, Date
 Volume*: 200 Number: 1 Publishing Date: March 2007 Pages*: 226-230 Number of VG Pages: Page Start: 226 Page End: 230 Sequence Number: DOI: 10.1016/j.cam.2005.12.016

 Note: (LaTeX) 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 $\log(1/s)$. URL for the Abstract: http://dx.doi.org/10.1016/j.cam.2005.12.016 Categories, Keywords: Descartes' Rule of Signs, Descartes-Jacobi Rule, Bernstein basis, root isolation, root separation HyperLinks / References / URLs: Copyright Message: 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. Personal Comments: Download Access Level: Internal

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BibTeX Entry:

@ARTICLE{Eigenwillig2007a,
AUTHOR = {Eigenwillig, Arno},
TITLE = {On Multiple Roots in {Descartes'} Rule and Their Distance to Roots of Higher Derivatives},
JOURNAL = {Journal of Computational and Applied Mathematics},
PUBLISHER = {Elsevier},
YEAR = {2007},
NUMBER = {1},
VOLUME = {200},
PAGES = {226--230},