Title: Geometry of a Class of Generalized Cubic Polynomials
Author(s): Christopher Frayer
Pages: 93-99
Cite as:
Christopher Frayer, Geometry of a Class of Generalized Cubic Polynomials, Int. J. Anal. Appl., 8 (2) (2015), 93-99.

Abstract


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.

Full Text: PDF

 

References


  1. Christopher Frayer, Myeon Kwon, Christopher Schafahuser, and James A. Swenson, The Geometry of Cubic Polynomials, Math. Magazine 87 (2014), no. 2, 113–124.

  2. Dan Kalman, An elementary proof of Marden’s theorem, Amer. Math. Monthly 115 (2008), no. 4, 330–338.

  3. Sam Northshield, Geometry of Cubic Polynomials, Math. Magazine 86 (April 2013), 136–143.

  4. E.B Saff and A.D Snider, Fundamentals of Com,plex Analysis for Mathematics, Science, and Engineering, Prentice-Hall, Anglewood Cliffs, New Jersey, 1993.