Let E denote a vector space
over an algebraically closed field K of characteristic zero. Our object is to investigate
the location of null-sets of generalized polars of the product of certain given abstract
homogeneous polynomials from E to K. Some special aspects of this general problem
were studied in the complex plane by Bôcher and Walsh and, later, in vector spaces
by Marden. Our present treatment furnishes further generalizations of the theorems
of Marden, Bôcher, and Walsh and offers a systemmatic, abstract, and unified
approach to their completely independent methods. One of our results, in special
setting, relates to the polar of a product and reduces essentially to the author’s
earlier generalization [Trans. Amer. Math. Soc., 218 (1976), 115–131] of
Hörmander’s theorem on polars of abstract homogeneous polynomials. We
show also that our theorems cannot be further generalized in certain natural
directions.
