The Number of Complex Roots of a Univariate Polynomial Within a Rectangle

DOI: 10.5176/2251-1911_CMCGS14.03

Authors: Ganhewalage Jayantha Lanel, Charles Ching-An Cheng

Abstract: 

Let f (z) be a nonzero complex univariate polynomial and let R be a rectangle in the complex plane. The number of complex roots of f (z) inside R is given by the winding number of f (z) on R if f (z) has no roots on the boundary of R. In this paper the result is extended to all rectangles R even when f (z) has roots on the boundary of R under the condition that f (z) is square-free. It can also be used to formulate an algorithm that isolates the complex roots of any polynomial.

Keywords: polynomial; real root isolation; complex root isolation.