Grobner bases and the Nagata automorphism

We study different properties of the Nagata automorphism of the polynomial algebra in three variables and extend them to other automorphisms of polyno mial algebras and algebras close to them. In particular, we propose two app roaches to the Nagata conjecture: via the theory of Grobner bases and tryin g to lift the Nagata automorphism to an automorphism of the free associativ e algebra. We show that the reduced Grobner basis of three face polynomials of the Nagata automorphism obtained by substituting a variable by zero doe s not produce an automorphism, independently of the "tag" monomial ordering , contrary to the two variable case. We construct examples related to Nagat a's automorphism which show different aspects of this problem. We formulate a conjecture which implies Nagata's conjecture. We also construct an expli cit lifting of the Nagata automorphism to the free metabelian associative a lgebra. Finally, we show that the method to determine whether an endomorphi sm 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. (C) 1999 Elsevier Science B.V. All rights reser ved.