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Changing the variables in the Bethe Ansatz Equations (BAE) for the XXZsix-vertex model we had obtained a coupled system of polynomial equations. Thisprovided a direct link between the BAE deduced from the Algebraic Bethe Ansatz(ABA) and the BAE arising from the Coordinate Bethe Ansatz (CBA). For twomagnon states this polynomial system could be decoupled and the solutions givenin terms of the roots of some self-inversive polynomials. From theoremsconcerning the distribution of the roots of self-inversive polynomials we madea thorough analysis of the two magnon states, which allowed us to find thelocation and multiplicity of the Bethe roots in the complex plane, to discussthe completeness and singularities of Bethe's equations, the ill-foundedstring-hypothesis concerning the location of their roots, as well as to find aninteresting connection between the BAE with Salem's polynomials.

Where are the roots of the Bethe Ansatz equations?