Bethe roots for periodic TASEP and algebraic curve

We present an algebraic method for solving the Bethe ansatz equations for the periodic totally asymmetric exclusion process (TASEP) with an arbitrary number of sites and particles. The Bethe ansatz equations are realized as an algebraic equation on a certain Riemann surface. While our Riemann surface is essentially the same as the one introduced by Prolhac, we focus on its algebraic realization as a (singular) plane curve. Through a counting argument on the Riemann surface, we establish a rigorous proof that the Bethe ansatz equation has the expected number of solutions when counted with multiplicity. Consequently, under appropriate generic conditions, the completeness of the Bethe ansatz follows. The decomposition of the Riemann surface into connected components determines how often each value of the product of Bethe roots appears. We classify the connected components and their multiplicities using a similar argument to the spectral degeneracy of the Markov matrix discussed by Golinelli-Mallick. As a result, we give an algebro-geometric characterization of Golinelli-Mallick-type spectral degeneracy of the Markov matrix. We also give explicit formulas for the number of connected components, the number of ramification points, and the total genus of the Riemann surface. These formulas recover the table of examples presented by Prolhac. Moreover, we explore applications of the special type of Bethe roots that appear in this case to partition functions of the five-vertex model. We introduce a version of the free energy and evaluate the thermodynamic limit to find the explicit form in terms of the Riemann zeta function.