Locating Unimodular Roots

Further explorations of unimodular roots

“…We prove this by first examining a general nth degree polynomial. This would be useful if one wanted to extend this work in the direction indicated in ( [1], Problems for Investigation 2).…”

“…In [1], the authors define a (complex) root of a polynomial to be unimodular if it lies on the unit circle in the complex plane or is, equivalently, a complex number of the form e i for 2 R. They then give a complete classification of the pairs .n; k/ for which the polynomial z n C z k 1 has a unimodular root. Moreover, they determine the exact locations of the unimodular roots for each such pair .n; k/.…”

Location of the Zeros of Certain Complex‐Valued Harmonic Polynomials

“…e location of the zeros of analytic polynomials has been studied by many researchers (see Brilleslyper and Schaubroeck [1], Dhemer [2], Frank [3], Gilewicz and Leopold [4], Howell and Kyle [5], Johnson and Tucker [6], Kennedy [7], and Melman [8]). Recently, researching the number of the zeros of general analytic trinomials and the regions in which the zeros are located has become of interest due to their application in other elds.…”

Location of the Zeros of Certain Complex-Valued Harmonic Polynomials

“…The location of the zeros of analytic polynomials has been studied by many researchers and we refer the reader to Brilleslyper and Schaubroeck [1], Dhemer [2], Frank [3], Gilewicz and Leopold [4], Howell and Kyle [5], Johnson and Tucker [6], Kennedy [7], and Melman [8]. Recently, researching the number of the zeros of general analytic trinomials and the regions in which the zeros are located has become of interest due to their application in other fields.…”

“…Consider p c (z) = z 5 + cz 3 − 1. Here, if n = 5 and k = 3, then n − k = 2.When c =1 2 , corresponding to the case 0 < c < 1, 1 2 and 3 2 are the positive real roots of the equationsx 2 + 1 2 − 1 = x 2 − 1 2 = 0 and x 2 − 1 2 − 1 = x 2 − 3 2 = 0 respectively. Therefore all the zeros of p c (z) = z 5 + 1 2 z 3 − 1 are contained in 1 2 ≤ |z| ≤ 3 2 .…”