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Zeros of Polar‐Composite Polynomials in Algebraically Closed Fields
Author(s) -
Zaheer Neyamat,
Alam Mahfooz
Publication year - 1980
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/s3-40.3.527
Subject(s) - algebraically closed field , mathematics , laguerre polynomials , complex plane , polynomial , plane (geometry) , zero (linguistics) , pure mathematics , class (philosophy) , polar , field (mathematics) , composite number , type (biology) , polar coordinate system , algebra over a field , mathematical analysis , geometry , algorithm , linguistics , philosophy , ecology , physics , astronomy , artificial intelligence , computer science , biology
Various composite polynomials have been studied as to the location of their zeros by a number of mathematicians, but mostly in relation to the complex plane. In more general spaces, however, this problem has so far been studied by Zervos, Marden, and Zaheer, but only for a special type of composite polynomial. In this paper, we introduce the class of composite polynomials L u (z) (respectively L v * (z) ), derived from a given polynomial and its polar (respectively formal) derivatives in an algebraically closed field K of characteristic zero, and we investigate the location of the zeros of such polynomials. A comparative study of the two classes shows that they are essentially the same and differ only in form. From our main theorem we deduce as corollaries some known results due to Laguerre, Zervos, and Zaheer. It may be noted that although all our results are obviously valid in the complex plane, we mention only those applications which are interesting from a geometrical standpoint or which have physical interpretations.