Eulerian operators and the Jacobian conjecture

Abstract

In this paper we introduce a new class of polynomial maps, the so-called nice polynomial maps. Using Eulerian operators we show how for these polynomial maps the main results obtained by Bass (Differential structure of étale extensions of polynomial algebras, Proc. Workshop on Commutative Algebra, MSRI, 1987) can be proved in a very simple and elementary way. Furthermore we show that every polynomial map F F satisfying the Jacobian condition, det J F ∈ k ∗ JF \in {k^{\ast }} , is equivalent to a nice polynomial map; more precisely the polynomial map F ( λ ) ( X ) = F ( X + λ ) − F ( λ ) {F_{(\lambda )}}(X) = F(X + \lambda ) - F(\lambda ) is nice for almost all λ ∈ k n \lambda \in {k^n} .