On the Jacobian conjecture: a new approach via Grobner bases

Abstract

In this paper we propose a new approach to the Jacobian conjecture via the theory of Grobner bases. We give a Grobner base criterion for a polynomial map to be polynomially invertible. Using this and using a result of Bass concerning the inverse degrees of automorphisms of polynomial rings, we reduce the Jacobian conjecture to certain problems in the complexity theory of Grobner bases. As a by-product, we construct examples which show the non-existence of a universal bound of degrees of Grobner bases over a polynomial ring in one variable over a field.