  Let K<X> be a free associative algebra over a field K of characteristic 0 and
let each of the noncommuting polynomials f,g generate its centralizer in K<X>.
Assume that the leading homogeneous components of f and g are algebraically
dependent with degrees which do not divide each other. We give a counterexample
to the recent conjecture of Jie-Tai Yu that deg([f,g])=deg(fg-gf) >
min{deg(f),deg(g)}. Our example satisfies deg(g)/2 < deg([f,g]) < deg(g) <
deg(f) and deg([f,g]) can be made as close to deg(g)/2 as we want. We obtain
also a counterexample to another related conjecture of Makar-Limanov and
Jie-Tai Yu stated in terms of Malcev - Neumann formal power series. These
counterexamples are found using the description of the free algebra K<X>
considered as a bimodule of K[u] where u is a monomial which is not a power of
another monomial and then solving the equation [u^m,s]=[u^n,r] with unknowns
r,s in K<X>.
