English
 
Help Privacy Policy Disclaimer
  Advanced SearchBrowse

Item

ITEM ACTIONSEXPORT

Released

Journal Article

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

MPS-Authors
/persons/resource/persons44369

Eigenwillig,  Arno
Algorithms and Complexity, MPI for Informatics, Max Planck Society;

External Resource
No external resources are shared
Fulltext (restricted access)
There are currently no full texts shared for your IP range.
Fulltext (public)
There are no public fulltexts stored in PuRe
Supplementary Material (public)
There is no public supplementary material available
Citation

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.


Cite as: https://hdl.handle.net/11858/00-001M-0000-000F-2019-9
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 $\log(1/s)$.