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Transactions of the American Mathematical Society

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On composite abstract homogeneous polynomials

Author: Neyamat Zaheer
Journal: Trans. Amer. Math. Soc. 228 (1977), 345-358
MSC: Primary 12D10; Secondary 30A08
MathSciNet review: 0429847
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Abstract: We study the null-sets of composite abstract homogeneous polynomials obtained from a pair of abstract homogeneous polynomials defined on a vector space over an algebraically closed field of characteristic zero. First such study for ordinary polynomials in the complex plane was made by Szegö, Cohn, and Egerváry and Szegö's theorem was later generalized to fields and vector spaces, respectively, by Zervos and Marden. Our main theorem in this paper further generalizes their results and, in the complex plane, improves upon Szegö's theorem and some other classical results. The method of proof is purely algebraic and utilizes the author's vector space analogue [Trans. Amer. Math. Soc. 218 (1976), 115-131] of Grace's theorem on apolar polynomials. We also show that our results cannot be further generalized in certain directions.

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Keywords: Abstract homogeneous polynomials and their polars, apolar polynomials, composite polynomials, circular cones, hermitian cones, generalized circular regions
Article copyright: © Copyright 1977 American Mathematical Society

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