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On the Parameterization of solutions for equations in Free Groups

In this paper we study the question of representing the set of solutions of an equation in a free group as the set of values of finitely many parametric functions depending on free word variables and, possibly, on certain other parameters. The main results says that if we allow in our representations arbitrary superpositions of arbitrary basic parametric functions having less than g free word variables, then the set of solutions of the quadratic coefficient-free equation x1x2…x2g=x2g…x2x1 (which is equivalent to [x1, y1]…[xg, yg]=1) cannot be represented in this way. A similar statement holds for the same equation considered over a free semigroup.