International Journal of Mathematics and Mathematical Sciences
Volume 10 (1987), Issue 4, Pages 757-776
doi:10.1155/S0161171287000851
Some theorems on generalized polars with arbitrary weight
1Mathematics Department, King Saud University, Riyadh, Saudi Arabia
2Mathematics Department, Aligarh Muslim University, Aligarh, India
Received 2 September 1986
Copyright © 1987 Neyamat Zaheer and Aijaz A. Khan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
The present paper, which is a continuation of our earlier work in Annali di
Mathematica [1] and Journal Math. Seminar [2] (EγEUθPIA), University of Athens,
Greece, deals with the problem of determining sufficiency conditions for the nonvanishing
of generalized polars (with a vanishing or nonvanishing weight) of the
product of abstract homogeneous polynomials in the general case when the factor
polynomials have been preassigned independent locations for their respective null-sets.
Our main theorems here fully answer this general problem and include in them,
as special cases, all the results on the topic known to date and established by Khan,
Marden and Zaheer (see Pacific J. Math. 74 (1978), 2, pp. 535-557, and the papers
cited above). Besides, one of the main theorems leads to an improved version of
Marden's general theorem on critical points of rational functions of the form
f1f2…fp/fp+1…fq, fi being complex-valued polynomials of degree ni.