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

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.