Degree estimate for commutators
| Autor: | Vesselin Drensky, Jie-Tai Yu |
|---|---|
| Rok vydání: | 2009 |
| Předmět: |
Coordinates
Automorphisms Wild automorphisms Field (mathematics) 16Z05 Degree estimate 16S10 16W20 K[u]-bimodules Jacobian conjecture Free algebra Tame automorphisms FOS: Mathematics Free associative algebras Mathematics Discrete mathematics Algebra and Number Theory Degree (graph theory) Formal power series Mathematics - Rings and Algebras Commutators Centralizer and normalizer Rings and Algebras (math.RA) Polynomial algebras Bimodule Equation [um s]=[un r] Jacobian Counterexample |
| Zdroj: | Journal of Algebra. 322:2321-2334 |
| ISSN: | 0021-8693 |
| DOI: | 10.1016/j.jalgebra.2009.07.018 |
| Popis: | Let K 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. 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 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. Comment: 18 pages |
| Databáze: | OpenAIRE |
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