The location of the zeros of the higher order derivatives of a polynomial

Author:
Piotr Pawlowski

Journal:
Proc. Amer. Math. Soc. **127** (1999), 1493-1497

MSC (1991):
Primary 30C15; Secondary 65E05

DOI:
https://doi.org/10.1090/S0002-9939-99-04695-X

Published electronically:
February 4, 1999

MathSciNet review:
1476386

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Abstract: Let be a complex polynomial of degree having zeros in a disk . We deal with the problem of finding the smallest concentric disk containing zeros of . We obtain some estimates on the radius of this disk in general as well as in the special case, where zeros in are isolated from the other zeros of . We indicate an application to the root-finding algorithms.

**1.**D. Coppersmith, C.A. Neff,*Roots of a Polynomial and its Derivatives.*Proceedings of the Fifth Annual ACM-SIAM Symposium on Discrete Algorithms (Arlington, VA, 1994), 271-279, ACM, New York, 1994. MR**95c:30008****2.**M. Marden,*Geometry of Polynomials.*Math. Surveys 3, Amer. Math. Soc. Providence, R.I. 1966. MR**37:1562****3.**V.Y. Pan,*New Techniques for Approximating Complex Polynomial Zeros*Proceedings of the Fifth Annual ACM-SIAM Symposium on Discrete Algorithms (Arlington, VA, 1994), 260-270, ACM, New York, 1994. MR**95g:65194****4.**V.Y. Pan,*Sequential and Parallel Complexity of Approximate Evaluation of Polynomial Zeros*Computers and Math. (with Applications), 14 (1987) 8, 591-622. MR**88j:65101****5.**J. Renegar,*On the worst-case arithmetic complexity of approximating zeros of polynomials.*Journal of Complexity, 3 (1987) 90-113. MR**89a:68107****6.**A. Sch\H{o}nhage,*Equation Solving in Terms of Computational Complexity*, Proceedings of the International Congress of Mathematicians, Berkeley, California, 1986. MR**89h:68071****7.**S. Smale,*Newton's method estimates from data at one point*, The Merging Disciplines: New Directions in Pure, Applied and Computational Mathematics. 185-196, Springer-Verlag, 1986. MR**88e:65076**

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Additional Information

**Piotr Pawlowski**

Affiliation:
Department of Mathematics and Computer Science, Kent State University, Kent, Ohio 44242

Address at time of publication:
Summit Systems, Inc., 22 Cortlandt Street, New York, New York 10007

Email:
ppawlows@mcs.kent.edu, piotr_pawlowski@summithq.com

DOI:
https://doi.org/10.1090/S0002-9939-99-04695-X

Received by editor(s):
February 5, 1997

Received by editor(s) in revised form:
September 3, 1997

Published electronically:
February 4, 1999

Communicated by:
Theodore W. Gamelin

Article copyright:
© Copyright 1999
American Mathematical Society