On the location of critical points of polynomials

Authors:
Branko Curgus and Vania Mascioni

Journal:
Proc. Amer. Math. Soc. **131** (2003), 253-264

MSC (2000):
Primary 30C15; Secondary 26C10

DOI:
https://doi.org/10.1090/S0002-9939-02-06534-6

Published electronically:
June 3, 2002

MathSciNet review:
1929045

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Given a polynomial of degree and with at least two distinct roots let . For a fixed root we define the quantities and . We also define and to be the corresponding minima of and as runs over . Our main results show that the ratios and are bounded above and below by constants that only depend on the degree of . In particular, we prove that , for any polynomial of degree .

**[1]**P. Borwein, T. Erdélyi,*Polynomials and polynomial inequalities*, Graduate Texts in Mathematics, 161, Springer-Verlag, 1995. MR**97e:41001****[2]**P. Henrici,*Applied and computational complex analysis, Vol. 1*, John Wiley & Sons, 1988. MR**90d:30002****[3]**M. Marden,*Geometry of polynomials*, Second edition reprinted with corrections, American Mathematical Society, Providence, 1985. MR**37:1562****[4]**M. Mignotte,*Mathematics for computer algebra*, Springer-Verlag, 1992. MR**92i:68071****[5]**A. M. Ostrowski,*On the moduli of zeros of derivatives of polynomials*, J. Reine Angew. Math. 230 (1968), 40-50. MR**37:1563****[6]**J. L. Walsh,*On the location of the roots of the derivative of a polynomial*, C. R. Congr. Internat. des Mathématiciens, pp. 339-342, Strasbourg, 1920.**[7]**J. L. Walsh,*The location of critical points of analytic and harmonic functions*, Amer. Math. Soc. Colloq. Publ., Vol. 34, 1950. MR**12:249d**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
30C15,
26C10

Retrieve articles in all journals with MSC (2000): 30C15, 26C10

Additional Information

**Branko Curgus**

Affiliation:
Department of Mathematics, Western Washington University, Bellingham, Washington 98225

Email:
curgus@cc.wwu.edu

**Vania Mascioni**

Affiliation:
Department of Mathematics, Western Washington University, Bellingham, Washington 98225

Email:
masciov@cc.wwu.edu

DOI:
https://doi.org/10.1090/S0002-9939-02-06534-6

Keywords:
Roots of polynomials,
critical points of polynomials,
separation of roots

Received by editor(s):
July 10, 2001

Received by editor(s) in revised form:
September 4, 2001

Published electronically:
June 3, 2002

Communicated by:
N. Tomczak-Jaegermann

Article copyright:
© Copyright 2002
American Mathematical Society