Let ℱ be the free group
generated by vaγiablesX,Y , 𝒞 be any given free group and W,g be elements of ℱ,𝒞,
respectively. Then x = (x,y) is a solution of the binary homogeneous equation W = g
if and only if W(x) = g, where x,y are elements of 𝒞. A general method is obtained
which will determine, for arbitrarily given W,g, the solution set of W = g.
Effectiveness is an essential requirement for the method. It may be noted that the
problem can also be regarded as the problem of finding embeddings of ℱ into
𝒞. The problem of solving an equation splits naturally into two parts: the
existence problem, to determine whether there are solutions; and secondly,
the characterization of the general solution. The existence problem here is
solved by showing that the equation has a solution if and only if at least
one of a finite set of identities has a solution. The set of identities may be
obtained in an effective (and quite practical) way from W,g. In order to
solve the second part of the problem, it is necessary to analyse the way in
which a solution of an identity generates solutions of the equation. This is
elucidated by the introduction of a set of mappings Φ(W), the members of which
are (derived from) the automorphisms of ℱ. The members of Φ(W) serve
as parameters for the general solution. It has not been possible to specify
Φ(W) in an effective way (at least, not according to one interpretation of this
requirement), but Φ(W) is a group, and this fact can be used in applications of the
theory.
