The Jacobian conjecture was proposed by O.H. Keller in 1939. It asks whether a polynomial endomorphism of ℂn whose Jacobian is constant must be invertible. Despite its simple and reasonable statement, the conjecture has not been proved even in the two dimensional case. In this chapter we show that this conjecture would follow if one could prove that every endomorphism of the Weyl algebra is an automorphism. The chapter opens with a discussion of polynomial maps, which will play a central rôle in the second part of the book. We shall return to the Jacobian conjecture in Ch. 19.

POLYNOMIAL MAPS.

Let F : Kn → Km be a map and p a point of Kn. We say that F is polynomial if there exist F1, …, Fm ∈ K[x1, …, xn] such that F(p) = (F1(p), …, Fm(p)). A polynomial map is called an isomorphism or a polynomial isomorphism if it has an inverse which is also a polynomial map. It is not always the case that a bijective polynomial map has an inverse which is also polynomial. For an example where this does not occur see Exercise 5.1. However, if K = ℂ, every invertible polynomial map has a polynomial inverse. This is proved in [Bass, Connell and Wright; Theorem 2.1].