On the zeros of a polynomial and its derivatives

Author:
Piotr Pawlowski

Journal:
Trans. Amer. Math. Soc. **350** (1998), 4461-4472

MSC (1991):
Primary 30C15

DOI:
https://doi.org/10.1090/S0002-9947-98-02291-0

MathSciNet review:
1473453

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: If is univariate polynomial with complex coefficients having all its zeros inside the closed unit disk, then the Gauss-Lucas theorem states that all zeros of lie in the same disk. We study the following question: what is the maximum distance from the arithmetic mean of all zeros of to a nearest zero of ? We obtain bounds for this distance depending on degree. We also show that this distance is equal to for polynomials of degree 3 and polynomials with real zeros.

**1.**B. Anderson,*Polynomial Root Dragging*, Amer. Math. Monthly**100**(1993), 864-866. MR**94i:26006****2.**M. Marden,*Conjectures on the critical points of a polynomial*, Amer. Math. Monthly**90**(1983), 267-276. MR**84e:30007****3.**M. Marden,*Geometry of Polynomials*, Math. Surveys 3, Amer. Math. Soc., Providence, R.I. 1966. MR**37:1562****4.**M.J. Miller,*Maximal polynomials and the Ilieff-Sendov conjecture*, Trans. Amer. Math. Soc.**321**(1990), 285-303. MR**90m:30007****5.**M.J. Miller,*Continuous independence and the Ilieff-Sendov conjecture*, Proc. Amer. Math. Soc.**115**(1992), 79-83. MR**92h:30012****6.**G.V. Milovanovic, D.S. Mitrinovic, and Th.M. Rassias,*Topics in polynomials : extremal problems, inequalities, zeros*, World Scientific, River Edge, NJ, 1994. MR**95a:30009****7.**D. Phelps and R.S. Rodriguez,*Some properties of extremal polynomials for the Ilieff conjecture*, Kodai Math. Sem. Report**24**(1972), 172-175. MR**46:3753****8.**R. M. Robinson,*On the span of derivatives of polynomials*, Amer. Math. Monthly**71**(1964), 504-508. MR**28:5155****9.**R. M. Robinson,*Algebraic equations with span less than 4*, Math. Comp.**18**(1964), 547-559. MR**29:6624****10.**G. Schmeisser,*On Ilieff's Conjecture*, Math. Z.**156**(1977), 165-173. MR**58:6182**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (1991):
30C15

Retrieve articles in all journals with MSC (1991): 30C15

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 Cortland St., New York, New York 10007

Email:
piotr-pawlowski@summithq.com

DOI:
https://doi.org/10.1090/S0002-9947-98-02291-0

Keywords:
Polynomials,
location of zeros

Received by editor(s):
June 27, 1996

Article copyright:
© Copyright 1998
American Mathematical Society