Vol. 74, No. 2, 1978

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On generalized polars of the product of abstract homogeneous polynomials

Neyamat Zaheer

Vol. 74 (1978), No. 2, 535–557

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.

Mathematical Subject Classification 2000
Primary: 12D10
Received: 15 February 1977
Published: 1 February 1978
Neyamat Zaheer