Geometry of a Class of Generalized Cubic Polynomials

Geometry of a Class of Generalized Cubic Polynomials

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This paper studies a class of generalized complex cubic polynomials of the form p(z)=(z-1)(z-r_1)^k(z-r_2)^k where r_1 and r_2 lie on the unit circle and k is a natural number.  We completely characterize where the nontrivial critical points of p can lie, and to what extent they determine the polyno...

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Main Author: Christopher Frayer
Format: Article
Language:English
Published: Etamaths Publishing 2015-08-01
Series:International Journal of Analysis and Applications
Online Access:http://etamaths.com/index.php/ijaa/article/view/544
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spelling doaj-a8ed8c1ee87d4faeb7fe589f282d8c432021-08-26T13:44:36ZengEtamaths PublishingInternational Journal of Analysis and Applications2291-86392015-08-01829399132Geometry of a Class of Generalized Cubic PolynomialsChristopher Frayer0University of Wisconsin-PlattevilleThis paper studies a class of generalized complex cubic polynomials of the form p(z)=(z-1)(z-r_1)^k(z-r_2)^k where r_1 and r_2 lie on the unit circle and k is a natural number.  We completely characterize where the nontrivial critical points of p can lie, and to what extent they determine the polynomial.   The main results include (1) a nontrivial critical point of such a polynomial almost always determines the polynomial uniquely, and (2) there is a `desert' in the unit disk in which critical points cannot occur.http://etamaths.com/index.php/ijaa/article/view/544
collection DOAJ
language English
format Article
sources DOAJ
author Christopher Frayer
spellingShingle Christopher Frayer
Geometry of a Class of Generalized Cubic Polynomials
International Journal of Analysis and Applications
author_facet Christopher Frayer
author_sort Christopher Frayer
title Geometry of a Class of Generalized Cubic Polynomials
title_short Geometry of a Class of Generalized Cubic Polynomials
title_full Geometry of a Class of Generalized Cubic Polynomials
title_fullStr Geometry of a Class of Generalized Cubic Polynomials
title_full_unstemmed Geometry of a Class of Generalized Cubic Polynomials
title_sort geometry of a class of generalized cubic polynomials
publisher Etamaths Publishing
series International Journal of Analysis and Applications
issn 2291-8639
publishDate 2015-08-01
description This paper studies a class of generalized complex cubic polynomials of the form p(z)=(z-1)(z-r_1)^k(z-r_2)^k where r_1 and r_2 lie on the unit circle and k is a natural number.  We completely characterize where the nontrivial critical points of p can lie, and to what extent they determine the polynomial.   The main results include (1) a nontrivial critical point of such a polynomial almost always determines the polynomial uniquely, and (2) there is a `desert' in the unit disk in which critical points cannot occur.
url http://etamaths.com/index.php/ijaa/article/view/544
work_keys_str_mv AT christopherfrayer geometryofaclassofgeneralizedcubicpolynomials
_version_ 1721193628474277888

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