Functions for the general solution of parametric word equations

Abstract

In this article we introduce the functions ${}^{Fi}(x_1 ,x_2 )^{\lambda _1 ,...,\lambda _3 }$and ${}^{Th}(x_1 ,x_2 ,x_3 )^{\lambda _1 ,...,\lambda _{2^3 } }$(i=1,2,3), of the word variables x _i and of the natural number variables λ _i , where s ≥ 0. By means of these functions, we give exactly the general solution (i.e., the set of all the solutions) of the first basic parametric equation: x ₁ x ₂ x ₃ x ₄=x ₃ x ^λ₁ x ₂ x ₅, in a free monoid.