Polynomial-Time Complexity for Instances of the Endomorphism Problem in Free Groups

We say the endomorphism problem is solvable for an element W in a free group F if it can be decided effectively whether, given U in F , there is an endomorphism φ of F sending W to U . This work analyzes an approach due to C. Edmunds and improved by C. Sims. Here we prove that the approach provides an efficient algorithm for solving the endomorphism problem when W is a twogenerator word. We show that when W is a two-generator word this algorithm solves the problem in time polynomial in the length of U . This result gives a polynomial-time algorithm for solving, in free groups, two-variable equations in which all the variables occur on one side of the equality and all the constants on the other side.