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.
Highlights
A few years ago, the concept of generalized polars of the product of abstract homogeneous polynomials (a.h.p.) was introduced by Marden [3] while in his attempt to generalize to vector spaces a theorem due to Bcher [4]
His formulation involves the use of hermitian cones [5], a concept which was first used by HSrmander [6] in obtaining a vector space analogue of Laguerre’s theorem on polar-derivatives [7] and, later, employed by Marden [3], [8], in the theory of composite a.h.p.’s
All have a common feature that the factor polynomials involved in the generalized polar of the product have been divided into two or three groups, each of which is preassigned a circular cone containing the null-sets of all polynomials belonging to that group
Summary
A few years ago, the concept of generalized polars of the product of abstract homogeneous polynomials (a.h.p.) was introduced by Marden [3] while in his attempt to generalize to vector spaces a theorem due to Bcher [4]. His formulation involves the use of hermitian cones [5], a concept which was first used by HSrmander [6] in obtaining a vector space analogue of Laguerre’s theorem on polar-derivatives [7] and, later, employed by Marden [3], [8], in the theory of composite a.h.p.’s In all these areas the role of the class of hermitian cones has been replaced by a strictly larger class of the so-called circular cones. We shall call P an a. h. p. (resp. an algebra-valued a. h. p. if V is taken as K (resp. an algebra)
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
