Solving One-Variable Equations in Free Groups
  Equations in free groups have become prominent recently in connection with the solution to the well known Tarski Conjecture. Results of Makanin and Rasborov show that solvability of systems of equations is decidable and there is a method for writing down in principle all solutions. However, no practical method is known; the best estimate for the complexity of the decision procedure is P-space.   The special case of one variable equations in free groups has been open for a number of years, although it is known that the solution sets admit simple descriptions. We use cancellation arguments to give a short and direct proof of this result and also to give a practical polynomial time algorithm for finding solution sets. One variable equations are the only general subclass of equations in free groups for which such results are known.   We improve on previous attempts to use cancellation arguments by employing a new method of reduction derived from techniques from formal language theory. Our paper is self contained; we assume only knowedge of basic facts about free groups. 
Dimitri Bormotov, Robert Gilman, Alexei Myasnikov