Parametric equations in free groups (Q1916659)

If \(X(x_1,x_2,\dots,x_n)\) is a non-trivial free word in the variables \(x_1,x_2,\dots,x_n\) (and their inverses), then in the free group \(F_\infty=\langle a_1,a_2,\dots\rangle\) a coefficient free equation is of the form \(X(x_1,x_2,\dots,x_n)=1\). A solution of this equation is a set of non-trivial elements \(X_1,X_2,\dots,X_n\) of \(F_\infty\) such that \(X(X_1,\dots,X_n)\) is the identity in the free group \(F_\infty\). In this paper the author introduces the notion of parametric equations, which contains natural parameters as exponents and a system of linear Diophantine equations relating these exponents. Then in terms of orientable graphs, he introduces elementary transformations, which he calls graph substitutions for the description of general solutions of ordinary equations in a free group. He proves that it is possible to linearize any relation among parameters that appears in the course of transformation of the given equation. Detailed results are too technical to be stated here.