Certain applications of the theory of polar-composite polynomials
Authors:
Neyamat Zaheer and Mahfooz Alam
Journal:
Proc. Amer. Math. Soc. 85 (1982), 383-388
MSC:
Primary 30C15
DOI:
https://doi.org/10.1090/S0002-9939-1982-0656108-8
MathSciNet review:
656108
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Abstract: In a recent paper [5] the authors have, for the first time, given a detailed account of the theory of polar-composite polynomials in algebraically closed fields of characteristic zero. In another paper [6], we have given some applications of this theory and have obtained a few results for a new variety of composite polynomials which have been derived from certain polar-composite polynomials through iteration. In the present paper also we consider the same variety of composite polynomials, but our present study deals with a different aspect of application of the said theory. Besides other things, our main theorem here offers a generalization of a result due to Marden [2] (see also [1, Theorem (16,3)]).
- [1] Morris Marden, Geometry of polynomials, Second edition. Mathematical Surveys, No. 3, American Mathematical Society, Providence, R.I., 1966. MR 0225972
- [2] Morris Marden, The zeros of certain composite polynomials, Bull. Amer. Math. Soc. 49 (1943), 93–100. MR 0007809, https://doi.org/10.1090/S0002-9904-1943-07856-1
- [3] N. Zaheer, Null-sets of abstract homogeneous polynomials in vector spaces, Doctoral thesis, Univ. of Wisconsin, Milwaukee, 1971.
- [4] Neyamat Zaheer, On polar relations of abstract homogeneous polynomials, Trans. Amer. Math. Soc. 218 (1976), 115–131. MR 0401719, https://doi.org/10.1090/S0002-9947-1976-0401719-X
- [5] Neyamat Zaheer and Mahfooz Alam, Zeros of polar-composite polynomials in algebraically closed fields, Proc. London Math. Soc. (3) 40 (1980), no. 3, 527–552. MR 572018, https://doi.org/10.1112/plms/s3-40.3.527
- [6] Neyamat Zaheer and Mahfooz Alam, Some applications of the theory of polar-composite polynomials, J. London Math. Soc. (2) 22 (1980), no. 3, 403–410. MR 596319, https://doi.org/10.1112/jlms/s2-22.3.403
- [7] Spiros P. Zervos, Aspects modernes de la localisation des zéros des polynomes d’une variable, Ann. Sci. École Norm. Sup. (3) 77 (1960), 303–410 (French). MR 0125944
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Additional Information
DOI:
https://doi.org/10.1090/S0002-9939-1982-0656108-8
Keywords:
Generalized circular regions,
polar-composite polynomials
Article copyright:
© Copyright 1982
American Mathematical Society
