Summary: Changing the variables in the Bethe Ansatz Equations ({\mathsf {BAE}}) for the xxz six-vertex model we had obtained a coupled system of polynomial equations. This provided a direct link between the {\mathsf {BAE}} deduced from the Algebraic Bethe Ansatz ({\mathsf{FABA}}) and the {\mathsf {BAE}} arising from the Coordinate Bethe Ansatz ({\mathsf{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 {\mathsf{BAE}} with Salem’s polynomials.