Some theorems on generalized polars with arbitrary weight

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 f 1 f 2 … f p / f p + 1 … f q , f i being complex-valued polynomials of degree n i .