The non-parametrizability of the word equation xyz=zvx: A short proof

Although Makanin proved the problem of satisfiability of word equations to be decidable, the general structure of solutions is difficult to describe. In particular, Hmelevskii proved that the set of solutions of xyz=zvx cannot be described using only finitely many parameters, contrary to the case of equations in three unknowns. In this paper we give a short, elementary proof of Hmelevskii's result.