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

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

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引用

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.


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