  Changing the variables in the Bethe Ansatz Equations (BAE) for the XXZ
six-vertex model we had obtained a coupled system of polynomial equations. This
provided a direct link between the BAE deduced from the Algebraic Bethe Ansatz
(ABA) and the BAE arising from the Coordinate Bethe Ansatz (CBA). For two
magnon states this polynomial system could be decoupled and the solutions given
in terms of the roots of some self-inversive polynomials. From theorems
concerning the distribution of the roots of self-inversive polynomials we made
a thorough analysis of the two magnon states, which allowed us to find the
location and multiplicity of the Bethe roots in the complex plane, to discuss
the completeness and singularities of Bethe's equations, the ill-founded
string-hypothesis concerning the location of their roots, as well as to find an
interesting connection between the BAE with Salem's polynomials.
