### 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.

#### Keywords

polynomial; real root isolation; complex root isolation

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