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

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**, 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**, 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**, 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**, 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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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