Gröbner bases and the Nagata automorphism

Abstract

We 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 Gröbner bases and trying to lift the Nagata automorphism to an automorphism of the free associative algebra. We show that the reduced Gröbner 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.