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TitleGröbner bases and the Nagata automorphism
Author(s) Vesselin Drensky, Jaime Gutierrez, Jie-Tai Yu
TypeArticle in Journal
AbstractWe study different properties of the Nagata automorphism of the polynomial algebra in three variables and extend them to other automorphisms of polynomial algebras and algebras close to them. In particular, we propose two approaches to the Nagata conjecture: via the theory of Grobner bases and trying to lift the Nagata automorphism to an automorphism of the free associative algebra. We show that the reduced Grobner basis of three face polynomials of the Nagata automorphism obtained by substituting a variable by zero does not produce an automorphism, independently of the tag monomial ordering, contrary to the two variable case. We construct examples related to Nagata's automorphism which show different aspects of this problem. We formulate a conjecture which implies Nagata's conjecture. We also construct an explicit lifting of the Nagata automorphism to the free metabelian associative algebra. Finally, we show that the method to determine whether an endomorphism of K[X] is an automorphism is based on a general fact for the ideals of arbitrary free algebras and works also for other algebraic systems such as groups and semigroups, etc.
URL dx.doi.org/10.1016/S0022-4049(97)00146-1
JournalJournal of Pure and Applied Algebra
Translation No
Refereed No