Degree Estimate for Commutators

Let K〈X〉 be a free associative algebra over a field K of characteristic 0 and let each of the noncommuting polynomials f, g ∈ K〈X〉 generate its centralizer. 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 1 2 deg(g) < 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 MakarLimanov 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 [um, s] = [un, r] with unknowns r, s ∈ K〈X〉.