Universal Journal of Mathematics and Mathematical Sciences
Volume 10, Issue 1, Pages 49 - 68
(June 2017) http://dx.doi.org/10.17654/UM010010049 |
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A TOUR OF EXTREMES: THE GEOMETRY OF A FAMILY OF POLYNOMIALS WITH THREE ROOTS
Christopher Frayer
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Abstract: We study the critical points of complex-valued polynomials of the form with and The Gauss-Lucas theorem guarantees that the critical points of such a polynomial will lie within the unit disk. This paper further explores the location and structure of these critical points. For example, there is a ‘desert’, the open disk in which the critical points cannot occur. Furthermore, a critical point of such a polynomial almost always determines the polynomial uniquely. |
Keywords and phrases: geometry of polynomials, critical points, Marden’s theorem, Gauss-Lucas theorem. |
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