Geometry of a Class of Generalized Cubic Polynomials
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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| Format: | Article |
| Language: | English |
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Etamaths Publishing
2015-08-01
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| Series: | International Journal of Analysis and Applications |
| Online Access: | http://etamaths.com/index.php/ijaa/article/view/544 |
| Summary: | 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. |
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| ISSN: | 2291-8639 |