  In this paper we study in detail the properties of the algebraic winding
number proposed in a paper by M. Eisermann with respect to complex root
counting in rectangles. We also propose a new algebraic winding number which
computes the number of complex roots of a polynomial in a rectangle under no
assumptions, including roots on edges or vertices with appropriate counting. We
extend both winding numbers to rational functions, obtaining then an algebraic
version of the argument principle for rectangles.

Algebraic Geometry