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

Pawlowski, Piotr

doi:10.1090/S0002-9939-99-04695-X

The location of the zeros of the higher order derivatives of a polynomial
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Proc. Amer. Math. Soc. 127 (1999), 1493-1497 Request permission

Abstract:

Let $p(z)$ be a complex polynomial of degree $n$ having $k$ zeros in a disk $D$. We deal with the problem of finding the smallest concentric disk containing $k-l$ zeros of $p^{(l)}(z)$. We obtain some estimates on the radius of this disk in general as well as in the special case, where $k$ zeros in $D$ are isolated from the other zeros of $p(z)$. We indicate an application to the root-finding algorithms.

References

Don Coppersmith and C. Andrew Neff, Roots of a polynomial and its derivatives, Proceedings of the Fifth Annual ACM-SIAM Symposium on Discrete Algorithms (Arlington, VA, 1994) ACM, New York, 1994, pp. 271–279. MR 1285172
Morris Marden, Geometry of polynomials, 2nd ed., Mathematical Surveys, No. 3, American Mathematical Society, Providence, R.I., 1966. MR 0225972
Victor Y. Pan, New techniques for approximating complex polynomial zeros, Proceedings of the Fifth Annual ACM-SIAM Symposium on Discrete Algorithms (Arlington, VA, 1994) ACM, New York, 1994, pp. 260–270. MR 1285171
V. Pan, Sequential and parallel complexity of approximate evaluation of polynomial zeros, Comput. Math. Appl. 14 (1987), no. 8, 591–622. MR 920483, DOI 10.1016/0898-1221(87)90186-6
James Renegar, On the worst-case arithmetic complexity of approximating zeros of polynomials, J. Complexity 3 (1987), no. 2, 90–113. MR 907192, DOI 10.1016/0885-064X(87)90022-7
Arnold Schönhage, Equation solving in terms of computational complexity, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986) Amer. Math. Soc., Providence, RI, 1987, pp. 131–153. MR 934220
Steve Smale, Newton’s method estimates from data at one point, The merging of disciplines: new directions in pure, applied, and computational mathematics (Laramie, Wyo., 1985) Springer, New York, 1986, pp. 185–196. MR 870648