Degree estimate for commutators

TL;DR

This paper provides counterexamples to conjectures about the degree of commutators in free associative algebras, showing that the degree can be less than previously expected and challenging existing assumptions.

Contribution

It constructs explicit counterexamples to conjectures on the degree of commutators in free associative algebras, using bimodule structures and formal power series.

Findings

Counterexamples to Yu's conjecture on deg([f,g])

Counterexamples to Makar-Limanov and Yu's conjecture

Degree of commutators can be arbitrarily close to half of deg(g)

Abstract

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>.