Geometry of a Class of Generalized Cubic Polynomials

Frayer, Christopher

Research Article

Geometry of a Class of Generalized Cubic Polynomials

Christopher Frayer¹

University of Wisconsin-Platteville

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.

How to Cite

C. Frayer, Geometry of a Class of Generalized Cubic Polynomials, Int. J. Anal. Appl. 8 (2015), 93–99.

BibTeX

@article{Frayer2015,
  author  = {Frayer, Christopher},
  title   = {Geometry of a Class of Generalized Cubic Polynomials},
  journal = {Int. J. Anal. Appl.},
  volume  = {8},
  year    = {2015},
  pages   = {93--99},
  doi     = {}
}

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.