  In this paper, we propose a new algebraic winding number and prove that it
computes the number of complex roots of a polynomial in a rectangle, including
roots on edges or vertices with appropriate counting. The definition makes
sense for the algebraic closure C = R[i] of a real closed field R, and the root
counting result also holds in this case. We study in detail the properties of
the algebraic winding number defined in [3] with respect to complex root
counting in rectangles. We extend both winding numbers to rational functions,
obtaining then algebraic versions of the argument principle for rectangles.
