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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

On polar relations of abstract homogeneous polynomials


Author: Neyamat Zaheer
Journal: Trans. Amer. Math. Soc. 218 (1976), 115-131
MSC: Primary 12D10; Secondary 30A08
DOI: https://doi.org/10.1090/S0002-9947-1976-0401719-X
MathSciNet review: 0401719
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Abstract: In this paper we generalize, to vector spaces over algebraically closed fields of characteristic zero, two well-known classical results due to Laguerre and Grace, concerning, respectively, the relative location of the zeros of a complex-valued polynomial and its polar-derivative and the relative location of the zeros of two apolar polynomials. Vector space analogues of their results were generalized, to a certain degree, by Hörmander, Marden, and Zervos. Our results in this paper further generalize their results and, in the complex plane, improve upon those of Laguerre and Grace. Besides, the present treatment unifies their completely independent approaches into an improved and more systematic and abstract theory. We have also shown that our results are best possible in the sense that they cannot be further generalized in certain directions.


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DOI: https://doi.org/10.1090/S0002-9947-1976-0401719-X
Keywords: Abstract homogeneous polynomials and their polars, apolar polynomials, circular cones, hermitian cones, generalized circular regions, generalized balls
Article copyright: © Copyright 1976 American Mathematical Society