Reference: Bethe Ansatz Equations Could someone show me good references to find solutions of the Bethe Ansatz Equations, for simple cases (using algebraic geometry or others interfaces with mathematics)? 
For example in the case of XXZ model (arxiv.org/abs/math-ph/0306002), but don't have enough knowledge in this topic, so I found few articles, then any good references are welcome. 
 A: In this paper you can find a numerical treatment of the Bethe equations for the isotropic spin-$\frac{1}{2}$ Heisenberg quantum spin chain with periodic boundary conditions. 
A conjecture on the number of solutions with pairwise distinct roots of these Bethe equations in terms of numbers of so-called singular (or exceptional) solutions is formulated. More concretely, the  motivation behind the discussion in the reference is the completeness of the Bethe equations solutions and the presence of so called singular / exceptional solutions. These can be physical or unphysical.
In order to check the conjecture proposed in the paper one needs to find all solutions with pairwise distinct roots of the Bethe equations. As the check is highly non trivial, two numerical methods -homotopy continuation and the Baxter T-Q equation- are discussed up to an upper bound in the number of quantum spins in the one dimensional quantum spin chain. 
The homotopy continuation method finds the solutions of a polynomial system (in this case the Bethe equations) starting with a second polynomial system  (the start system) and a set of nonsingular solutions of it. The methods deforms the start system and the corresponding solutions to the system under interest to determine the solutions of the latter.
The Baxter T-Q equation method uses the model integrability; all details are contained in section 4 loc. cit. 
